Topological Properties of the Set of Admissible Transformations of Measures

Topological Properties of the Set of Admissible Transformations of Measures

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Дата:2006
Автор: Gabriyelyan, S.S.
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Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2006
Назва видання:Журнал математической физики, анализа, геометрии
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Цитувати:Topological Properties of the Set of Admissible Transformations of Measures / S.S. Gabriyelyan // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 9-39. — Бібліогр.: 18 назв. — англ.

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spelling irk-123456789-1065792016-10-01T03:02:10Z Topological Properties of the Set of Admissible Transformations of Measures Gabriyelyan, S.S. 2006 Article Topological Properties of the Set of Admissible Transformations of Measures / S.S. Gabriyelyan // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 9-39. — Бібліогр.: 18 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106579 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
format Article
author Gabriyelyan, S.S.
spellingShingle Gabriyelyan, S.S.
Topological Properties of the Set of Admissible Transformations of Measures
Журнал математической физики, анализа, геометрии
author_facet Gabriyelyan, S.S.
author_sort Gabriyelyan, S.S.
title Topological Properties of the Set of Admissible Transformations of Measures
title_short Topological Properties of the Set of Admissible Transformations of Measures
title_full Topological Properties of the Set of Admissible Transformations of Measures
title_fullStr Topological Properties of the Set of Admissible Transformations of Measures
title_full_unstemmed Topological Properties of the Set of Admissible Transformations of Measures
title_sort topological properties of the set of admissible transformations of measures
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2006
url http://dspace.nbuv.gov.ua/handle/123456789/106579
citation_txt Topological Properties of the Set of Admissible Transformations of Measures / S.S. Gabriyelyan // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 1. — С. 9-39. — Бібліогр.: 18 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT gabriyelyanss topologicalpropertiesofthesetofadmissibletransformationsofmeasures
first_indexed 2025-07-07T18:42:53Z
last_indexed 2025-07-07T18:42:53Z
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2006, vol. 2, No. 1, pp. 9�39 Topological Properties of the Set of Admissible Transformations of Measures S.S. Gabriyelyan Kharkov National Technic University "KPI" 21 Frunze Str., Kharkov, 61002, Ukraine E-mail:gabrss@kpi.kharkov.ua Received September 2, 2004 Suppose a topological semigroupG acts on a topological spaceX . A trans- formation g 2 G is called an admissible (partially admissible, singular, equi- valent, invariant) transformation for � relative to � if �g � � (accordingly: �g 6? �, �g ? �, �g � �, �g = c � �), where �g(E) := �(g�1E). We denote its collection by A(�j�) (accordingly: AP (�j�), S(�j�), E(�j�), I(�j�)). It is shown that all these sets are Borel subsets of very bounded types. In par- ticular, A(�j�) is a GÆ�Æ-subset of G. If G is a Polish group, then A(�j�), E(�j�) and I(�j�) admit a Polish topology. Key words: topological G-space, Polish G-space, measure, admissible transformation, Borel type, t-ergodic measure. Mathematics Subject Classi�cation 2000: 28C99, 37A99. Let X be a measurable space and � a probability measure on X. A transfor- mation g : X ! X is called admissible for � if �g � � where �g = �g�1. Such transformations are important for the study of measures. Let X be a topological group. The simplest transformations of X are translations. Denote by A(�) the set of admissible translations of �. For example, admissible translations arise naturally in the theory of stochastic processes. T.S. Pitcher [14] has done the general de�nition of an admissible translation and the simplest properties of A(�) for measures which correspond to stochastic processes. In detail some algebraic and topological properties for admissible translations of measures were considered by A.V. Skorohod [18] for a Hilbert space and by Y. Okazaki [12] for a separable metric group. It turned out that the structure of � depends on the �volume� of A(�) sub- stantially. In the case X = R and [0;1) � A(�), A.V. Skorohod [17] has proved that � is absolutely continuous relative to the Lebesgue measure and its support is of the form [a;1). P.L. Brockett [4] has generalized this fact to the case of c S.S. Gabriyelyan, 2006 S.S. Gabriyelyan locally compact �-compact groups. Moreover, the famous Mackey�Weil theorem [11] asserts that if X is a standard Borel group and A(�) = X, then X admits a locally compact topology and � is mutually absolutely continuous with respect to Haar measure. Transformations which take � to its equivalent (to oneself), constitute an important special case. We denote the set of such transformations by E(�) (I(�)). Restricting our considerations to such transformations only, we obtain a classical object of study in Ergodic theory. Let X be a locally compact group. There exists a measure (the Haar measure) such that I(�) = X. This fact plays a key role in Harmonic analysis. At any case, I(�) (the group of invariance) is compact and plays an important role in arithmetic of probability measures (see history and details in [8]). Let X = T be the circle group and � a probability measure on T. E(�) can be viewed as a group of eigenvalues for a nonsingular dynamical system and belongs to the class of so-called �saturated� subgroups [1, 9]. This approach demonstrates an interesting interplay between Harmonic analysis and Ergodic theory. These results constitute the basement of study of A(�) and similar sets in more detail. In particular, some algebraic, measure theoretical properties, together with a Lebegsue-type decomposition of this set have been considered (see [7]). In this article we study topological properties of the set of admissible transformations. Let X = G be a separable metric group. Y. Okazaki [12] has shown that the sets E(�) and A(�) are Borel. Let us consider the group E(�). Two methods of proving that E(�) is Borel are known. The �rst introduces the strong operator topology on E(�) and shows that this topolgy is Polish (see [18, 1, 13]). In the �rst part of the article we generalize this fact to all Polish G-spaces. Note that in the general case (see Remark 2.3), E(�) is not complete in the strong operator topology and it is necessary to amplify it by the initial topology. At the end of the �rst part we present some applications of our results to t-ergodic measures, which generalize corresponding facts for Abelian groups ([5, 9, 13]). The second method was used by Y. Okazaki [12]. In the second part of the article we use the de Possel theorem to prove much more (see Theorem 3.1). In particular, for X = T we establish that E(�) is the set of the type GÆ�Æ . This give a restriction on E(�) as well as saturation. 1. Preliminaries and basic de�nitions Let (X;B) be a Borel space. We can assume without loss of generality that X is separable, i.e.: if x; y 2 X and x 6= y, then there exists E 2 B such that x 2 E 63 y. De�nition 1.1. A pair (G;X) is called a (semi)group of transformations if: 1) G is a (semi)group and X is a Borel space; 10 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures 2) the mapping g : (X;B) ! (X;B), x 7! g � x, is Borel for all g 2 G; (gh) � x = g � (h � x), and if e is the unit in G then e � x = x, 8g; h 2 G, 8x 2 X: Let G be a topological (semi)group and X a topological space. Then the (semi)group of transformations (G;X) is said to be topological if the mapping (g; x) 7! g � x is continuous. De�nition 1.2. Let (G;X) and (H;Y ) be a (semi)group. A pair (p; �), where p : G ! H is a homomorphism and � : X ! Y a Borel mapping, is called a morphism from (G;X) to (H;Y ) if p and � acts as follows �(g � x) = p(g) � �(x) ; 8g 2 G; 8x 2 X: If (G;X) and (H;Y ) are topological (semi)groups of transformations then p and � are supposed to be continuous. Hence the set of [topological] (semi)groups of transformations form a category. Let E 2 B. The image and the inverse image of E is denoted by g � E and g�1E respectively. If G is a group, then g �E is denoted simply by gE. Let M(X) be the set of all �nite Borel measures on X. The subset of all posi- tive measures is denoted byM+(X). A measure � 2M+(X) is called probabilistic if �(X) = 1. The Dirac mass at a point x is denoted by Æx. Let �; � 2 M(X). We write �� � if j�j is absolutely continuous relative to j�j, and � ? � if j�j and j�j are mutually singular. Equivalence � � � means that � � � and � � �. If � = �1 + �2 whith �1 ? �2, then �1 and �2 are called parts of �. Let � 2 M(X) and g 2 G. Denote by �g the measure on (X;B) determined by the relation �g(E) = �(g�1E); E 2 B: Then (�g)h(E) = �g(h �1E) = �(g�1h�1E) = �hg(E), i.e., (�g)h = �hg. Let �, � 2M(X). One can represent them in the form � = �1 + �2; � = �1 + �2; where �1 � �1; �2 ? �; �2 ? �: This decomposition is called the Lebesgue decomposition of measures � and �. Denote by d� d� the derivative of � with respect to �. Then d� d� = d�1 d�1 ; �1 � a.e. ; and d� d� = 0; (�2 + �2)� a.e. Denote by mG the left Haar measure of a locally compact group G. For a function f(x) we put: f+(x) = maxff(x); 0g, f�(x) = minff(x); 0g. Then f(x) = f+(x)� f�(x). This article is devoted to the study of topological properties of the sets which are determined in the following de�nitions. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 11 S.S. Gabriyelyan De�nition 1.3. Let � 2 M(X). A transformation g 2 G is called an admis- sible (partially admissible, singular, equivalent, invariant) transformation for � if �g � �(respectively: �g 6? �, �g ? �, �g � �, �g = �). Their set denoted by A(�) (respectively: AP (�), S(�), E(�), I(�)). Obviously I(�) � E(�) � A(�) � AP (�); AP (�) \ S(�) = ;; AP (�) [ S(�) = G: It is clear that, if G has a unit e, then e 2 I(�). The following de�nition is a natural generalization of the previous one. De�nition 1.4. Let �, � 2 M(X). A transformation g 2 G is called an admissible (partially admissible, singular, equivalent, invariant) transformation for � relative to � if �g � � (respectively: �g 6? �, �g ? �, �g � �, �g = c � �, where c = k�k=k�k). Their set denoted by A(�j�) (respectively: AP (�j�), S(�j�), E(�j�), I(�j�)). Evidently, the corresponding inclusions are true for these sets: I(�j�) � E(�j�) � A(�j�) � AP (�j�); AP (�j�) \ S(�j�) = ;; AP (�j�) [ S(�j�) = G: Clearly, if G is a group, then E(�j�) = E(j�jjj�j), A(�j�) = A(j�jjj�j), AP (�j�) = AP (j�jjj�j), S(�j�) = S(j�jjj�j). Thus we will often restrict our considerations to probability measures only. The case when X = G is a group makes a special interest. The following operators arise naturally Lg(x) = gx ; Rg(x) = xg�1 ; Cg(x) = gxg�1 = LgRg(x); 8x; g 2 X: These operators determine the left, right and conjugate actions of G on X. By default, the action of G on X is left, i.e. g � x = gx. De�nition 1.5. Let G = X be a group. The sets AP (�j�), S(�j�), A(�j�), E(�j�), I(�j�) relative to the left (right, conjugate) action of the group on itself is denoted with the subindex l (respectively r; c), i.e., APl(�j�); Sl(�j�); Ar(�j�); Ec(�j�) etc. Put At(�j�) = Al(�j�) \ [Ar(�j�)]�1, At(�) = Al(�) \ [Ar(�)] �1 etc. 12 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures We remark that for noncommutative groups, P.L. Brockett [4] and Y. Okazaki [12] used the term admissible translations for the elements of At(�). Set G� = G [ fidXg. Then G� is a semigroup with unit. Corresponding sets relative to G� are denoted by AP �(�j�), A�(�j�) etc. A set E is called G-invariant if g�1(E) = E for all g 2 G. De�nition 1.6. A measure � is called t-ergodic, if for all its nonzero parts � and �, there exist g, h 2 G� such that �g 6? � and � 6? �h. Evidently, � is t-ergodic if and only if j�j is t-ergodic. 2. The strong topology on AP (�) Let X be a separable metric space. Let G be a separable metric (semi)group which acts continuously on X. Let � and � be probability measures on X. For � 2 L1(�) let �g = �1 + �2 be the Lebesgue decomposition of �g relative to �, where �1 � � and �2 ? �. Put T�;g(�) = �1: Then T�;g is a linear contractive operator from L1(�) to L1(�). Now we de�ne the strong operator topology (strong topology, for short) on AP (�j�) (compare with [18, 1, 13]). De�nition 2.1. A sequence gn 2 AP (�j�) is called convergent to g 2 AP (�j�) in the strong topology if lim n!1 kT�;gn(�)� T�;g(�)k = 0; 8� 2 L1(�); (2.1) for the semigroup case, and, additionally to (2.1), lim n!1 kT �;g �1 n (�)� T�;g�1(�)k = 0; 8� 2 L1(�); (2.2) for the group case. De�nition 2.2. Let g, h 2 AP (�j�). Let f�ng and f�ng be a countable dense subset in L1(�) and L1(�) respectively. If G is a semigroup we put d(h; g) = 1X n=1 1 2n kT�;h(�n)� T�;g(�n)k 1 + kT�;h(�n)� T�;g(�n)k ; and if G is a group we put d(h; g) = 1X n=1 1 2n � kT�;h(�n)� T�;g(�n)k 1 + kT�;h(�n)� T�;g(�n)k + kT�;h�1(�n)� T�;g�1(�n)k 1 + kT�;h�1(�n)� T�;g�1(�n)k � : Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 13 S.S. Gabriyelyan Notice some simple properties. Proposition 2.1. 1. d(g; h) is a pseudometric on AP (�j�). 2. The topology which determined by d(g; h) and the strong topology are coin- cide. 3. Let I(g; h) = fx : g � x = h � xg. Put �g;h(E) := �(E \ (X n I(g; h))) ; �g;h(E) := �(E \ (X n I(g; h))): Then d(g; h) is a metric if and only if the set AP (�g;hj�) \ fg; hg (in the case if G is a group, one of the sets AP (�g;hj�) \ fg; hg or AP (�g;hj�) \ fg�1; h�1g) is not empty for all g; h 2 AP (�j�); g 6= h. 4. If d(g; h) is a metric, then AP (�j�) is a separable metric space. 5. If G is a group, then the mapping j : AP (�j�) ! AP (�j�), j(g) = g�1, is a homeomorphism. 6. If h and t are invertible, then the mapping i(g) = tgh�1 is homeomorphism from B(�j�) to B(�hj�t), where B(:) is one of the sets AP (:), A(:), E(:), I(:). P r o o f. We give the proof for the case when G is a group. 1., 2. Evidently. 3. Let g 2 AP (�g;hj�). Let (�g;h)g = �1 + �2 with �1 � �, �2 ? �, and � be the part of �g;h such that �g = �1. By hypothesis, we can chose x0 2 supp � and a neighborhood U of x0 such that g � U \ h � U = ;. Put � = �jU = (�g;h)jU and chose �n such that k� � �nk < 0; 1k�k. Then (1 + kT�;g(�n)� T�;h(�n)k)2nd(g; h) � kT�;g(�n)� T�;h(�n)k = kT�;g(�)� T�;h(�) + T�;g(�n � �)� T�;h(�n � �)k � kT�;g(�)� T�;h(�)k � 2k�n � �k � kT�;g(�)k � 0; 2k�k = 0; 8k�k > 0: Hence d(g; h) > 0. Analogically, if g�1 2 AP (�g;hj�), then d(g; h) > 0. Conversely. Let d be a metric and g 6= h. Then d(g; h) > 0. Let �n = �1n+�2n with �1n � �g;h, � 2 n ? �g;h. Then �2n is concentrated on I(g; h) and therefore (�2n)g = (�2n)h. If g; h 2 S(�g;hj�), then kT�;g(�n)� T�;h(�n)k = kT�;g(�1n)� T�;h(�1n) + T�;g(� 2 n)� T�;h(�2n)k = 0: Similarly, if g�1; h�1 2 S(�g;hj�), then kT�;g�1(�n) � T�;h�1(�n)k = 0: Thus d(g; h) = 0. This contradiction concludes the proof. 14 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures 4. For k; l 2 N and m = (m1; : : : ;ml);q = (q1; : : : ; ql) 2 Nl , we put Uk;l;m;q =� g 2 AP (�j�) : kT�;g(�n)� �mn k < 1 k ; kT�;g�1(�n)� �qnk < 1 k ; n = 1; : : : ; l � : Select one element in every nonempty set Uk;l;m;q. Then we get at most countable set R. It is easy to show that R is dense in AP (�j�). 5. By Theorem 4.2 [7], the mapping j is a bijection. Denote by d1 a pseudo- metric on AP (�j�) and the corresponding function on AP (�j�) denoted by d2. It follows from our construction that d2(g �1; h�1) = d1(g; h): Hence d2 is a pseudometric too, and the mapping j is a homeomorphism (of the spaces with pseudometrics). Notice that if d1(g; h) is a metric, then d2(g; h) is a metric too and j is a metric isomorphism. 6. By Theorem 4.2 [7], the mapping i is a bijection. It is clear that f(�n)hg and f(�n)tg forms a dense subset in L1(�h) and L1(�t) respectively. Denote the corresponding pseudometric on AP (�hj�t) by d1. Since T�t;tgh�1( (�n)h) = T�( (�n)g) = T�;g(�n) and T�h;hg�1t�1((�n)t) = T�h( (�n)hg�1) = T�( (�n)g�1) = T�;g�1(�n), then d1(tg1h �1; tg2h �1) = d(g1; g2): Hence i is a homeomorphism. In particular, if d is a metric, then d1 is a metric too, and i is an isometrics. In the following proposition we study continuity of algebraic operations and elementary topological properties of the sets A(�j�); E(�j�) and I(�j�) in the strong topology. Proposition 2.2. Let g, h 2 AP (�j�) 1. If gn 2 A(�j�) and d(gn; g)! 0, then g 2 A(�j�). If gn; hn 2 A(�); d(gn; g)! 0 and d(hn; h)! 0, then d(gnhn; gh)! 0. Let G be a group (this condition is essentially), then 2. If gn 2 E(�j�)[I(�j�)] and d(gn; g)! 0, then g 2 E(�j�)[I(�j�)]. 3. If gn; hn 2 E(�)[I(�)]; d(gn; g)! 0 and d(hn; h)! 0; then d(gnhn; gh)! 0. 4. If gn; g 2 E(�)[I(�)] and d(gn; g)! 0; then d(g�1 n ; g�1)! 0. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 15 S.S. Gabriyelyan In particular, if d(g; h) is a metric, then A(�j�) is closed in AP (�j�) in the strong topology. If, in addition, G is a group, then E(�j�) and I(�j�) is closed in AP (�j�) with respect to the strong topology; A(�) is a closed topological semigroup; E(�) and I(�) are closed topological groups. P r o o f. We prove the proposition assuming that G is a group. 1. Let gn 2 A(�j�) and gn ! g in the strong topology. Let �g = �1+�2 with �1 � �, �2 ? �. It is necessary to show that �2 = 0. In the converse case, let > 0 be a part of j�j such that g � �2. Then T�;g( ) = 0 and kT�;gk( )� T�;g( )k = kT�;gk( )k = k k 6! 0; which is a contradiction. Let gn ! g and hn ! h with respect to the strong topology. Then g; h 2 A(�). Let " > 0 and � 2 L1(�). Then �h 2 L1(�). Choose N such that kT�;hn(�) � T�;h(�)k = k�hn � �hk < 1 2 " and kT�;gnh(�) � T�;gh(�)k = k(�h)gn � (�h)gk < 1 2 " for all n > N . Then kT�;gnhn(�)� T�;gh(�)k = k�gnhn � �ghk � k�gnhn � �gnhk+ k�gnh � �ghk < " ; 8n > N: Hence lim n!1 kT�;gnhn(�) � T�;gh(�)k = 0: (2.3) Further, put � g �1 n � �g�1 = �1n + �2n with �1n � �; �2n ? �. By the hypothesis kT �;g �1 n (�)� T�;g�1(�)k = k�1nk ! 0: Assume that (�2n)h�1n = 1n + 2n with 1n � �; 2n ? �. Let Æn be the part of �2n such that (Æn)h�1n = 1n: Then Æn = ( 1n)hn � �. This contradicts to our choice of �2n. Whence 1n = 0 and kT �;h �1 n g �1 n (�)� T �;h �1 n g�1 (�)k = kT�((�g�1n � �g�1)h�1n )k = kT�((�1n)h�1n )k � k�1nk ! 0: Let �g�1 = �1 + �2 with �1 � �; �2 ? �. Analogously, we can prove that �2 h �1 n and �2 h�1 are mutually singular with �. Since �1 is not depended on n, then kT �;h �1 n g�1 (�) � T�;h�1g�1(�)k = kT�;h�1n (�1)� T�;h�1(�1)k ! 0: 16 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures Thus kT �;h �1 n g �1 n (�)� T�;h�1g�1(�)k � kT �;h �1 n g �1 n (�)� T �;h �1 n g�1 (�)k + kT �;h �1 n g�1 (�)� T�;h�1g�1(�)k ! 0: By the above and (2.3), we see that gnhn ! gh in the strong topology. 2. Let gn 2 E(�j�) and gn ! g in the strong topology. Then g 2 A(�j�). Moreover, by Proposition 2.1, g�1 n tends to g�1 with respect to the strong topology on AP (�j�). Since g�1 n 2 E(�j�), then g�1 2 A(�j�). Thus g 2 E(�j�). Let gn 2 I(�j�) and gn ! g in the strong topology. It is proved that g 2 E(�j�). Since lim n!1 kT�;gk(�)� T�;g(�)k = lim n!1 kc � � � �gk = 0; then �g = c � � and g 2 I(�j�). 3., 4. If gn ! g and hn ! h in the strong topology, then g, h 2 E(�)[I(�)] by item 2. By Proposition 2.1 (5) and item 1, g�1 n ! g�1 and gnhn ! gh with respect to the strong topology on E(�). To prove the main theorem of this section we need three lemmas as follows. Lemma 2.1. Let fgng � AP (�j�) be a fundamental sequence in the strong topology, gn tends to g with respect to the original topology, � � � and the following condition is ful�lled (i) lim n!1 kT�;gn(�)k = kT�;g(�)k: Then lim n!1 kT�;gn(�) � T�;g(�)k = 0: (2.4) P r o o f. We can assume without loss of generality that � > 0: Represent � in the form � = �n + n = � + ; with �ngn � �; ngn ? �; �g � �; g ? �: Let �ngn = fn� and �g = f�. Evidently, � �ngn is fundamental if and only if ffng is fundamental in L1(�). Hence there exist the limit a := limn!1 kT�;gn(�)k and a subsequence fnk which converges to F (x) �-a.e. Then a = kFk = kT�;g(�)k = kfk: Clearly, it is enough to prove (2.4) for a subsequence. Thus we can assume without loss of generality that nk = k. It is necessary to prove that F = f�-a.e. Since kFk = kfk, it is enough to prove that F (x) � f(x) � � a.e. (2.5) Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 17 S.S. Gabriyelyan In the converse case, there exists a compact K such that: �(K) > 0; F (x) > f(x) on K; g(K) = 0 and fn(x) converges uniformly to F (x) on K. Since �g(K) = �g(K), then for some " > 0 we can �nd a neighborhood W � g�1K such that �(W ) < �(g�1K) + " = Z K f(x)d� + " < Z K F (x)d� � ": Since gn ! g, we can �nd N such that g�1 n K �W and �n(g�1 n K) > Z K F (x)d� � "; 8n > N: Then Z K F (x)d� � " < �n(g�1 n K) � �(W ) < Z K F (x)d� � "; 8n > N: This contradiction concludes the proof. R e m a r k 2.1. Condition (i) is important. Really, let G = X = R and � = � = 1 3 (Æ0 + Æ2 + mj[�1;1]): Then gn = 2 � 1 n 2 AP (�j�), gn converges to g = 2 2 AP (�j�) in the original topology. It is easily be checked that fgng is fundamental in the strong topology and does not satisfy condition (i). It is obvious that (2.4) is fails. Lemma 2.2. Let G be a semigroup [group] and fgng � A(�j�)[E(�j�)] a fundamental sequence in the strong topology. If gn converges to g with respect to the initial topology, then g 2 A(�j�)[E(�j�)]: P r o o f. Assume the converse and g 62 A(�j�): Then there exists a part � with a compact support K of the measure � such that �g ? �: For every natural number n we choose an open setWn such that g �K �Wn and �gn(Wn) < 0; 1k�k. Put Kn = K n g�1 n Wn � K. Then Kn is compact and �(Kn) � �(K)� �(g�1 n Wn) > 0; 9k�k: Since the maps x 7! gk �x are continuous on K and converge to the map x 7! g �x, then there existsm > n such that gm�Kn �Wn. In particular, gm�Kn\gn�Kn = ;. Then kT�;gm(�) � T�;gn(�)k = k�gm � �gnk � (�gm � �gn)(gm �Kn) � �(Kn)� �gn(Wn) > 0; 8k�k: 18 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures This contradicts to the fact that fgng is fundamental. If fgng � E(�j�), then we have proved that g 2 A(�j�). By Propositions 2.1, 2.2 and Theorem 4.2 [7], fg�1 n g � E(�j�) is fundamental in the strong topology on E(�j�). Thus g�1 2 A(�j�): Hence g 2 E(�j�). Lemma 2.3. Let G be a semigroup [group], fgng � A(�j�)[E(�j�)] a funda- mental sequence in the strong topology, and suppose that gn converges to g with respect to the original topology. Then g 2 A(�j�)[E(�j�)] and gn converges to g in the strong topology. P r o o f. By Lemma 2.2, g 2 A(�j�)[E(�j�)]. Then kT�;gn(�)k = k�gnk = k�k = k�gk;8�� �. By Lemma 2.1, we have lim n!1 kT�;gn(�)� T�;g(�)k = 0 ; 8�� �; and the lemma is proved for the semigroup case. Let G be a group. It is remain to prove that (2.2) is true for all measures �, 0 < � � �, i.e. limn!1 kT�;g�1n (�)� T�;g�1(�)k = 0, Represent � in the form � = �n + n = �0 + 0 with �n g �1 n � �; n g �1 n ? �; �0 g�1 � �; 0 g�1 ? �: By hypothesis, �n g �1 n is fundamental and thus converges to some measure �� �. Hence �n = �gn + (�n � �gn); where k�n � �gnk = k�ng�1n � �k ! 0 and �gn ! �g (this follows from 2.1, since � � � and gn 2 A(�j�) ). In particular, �n ! �g. Since �n is the part of �, then �n = �n�, where �n takes part two values 0 and 1. Then �n converges to a function � on some subsequence. The function � takes part only values 0 and 1 too. Clearly that �g = ��. Thus �g is a part of �. From the proof of Lemma 2.1 (see (2.5)) it follows that �0 = � + �g, where � is a part of � and � ? �g: It is necessary to prove that � = 0, since, in this case, condition (i) of Lemma 2.1 is true. Let � 6= 0. Then �g�1 � �. Thus, by Lemmas 2.1 and 2.2, (�g�1)gn ! (�g�1)g = �: Choose and �x a natural number n so much that k�n � �gk < 0; 1k�k ; k�gng�1 � �k < 0; 1k�k: (2.6) Choose E1 and E2 such that �njE1 = �gjE1 ; �n(E2) < 0; 1k�k; �g(E2) = 0 and �n(X n (E1 [E2)) = 0: Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 19 S.S. Gabriyelyan If � = � + �1 with � � �gng�1 ; �1 ? �gng�1 , then (2.6) implies k�1k < 0; 1k�k. Since � and �n are parts of �, then, by our choice of E2, we have �jE2 = �njE2 and �(E2) � �n(E2). Thus �(X n (E1 [E2)) = �(X)� (�(E1) + �(E2)) = �(X)� �(E2) � �(X)� �n(E2) = �(X)� �1(X) � �n(E2) > k�k � 0; 2k�k = 0; 8k�k: By the above there exists a part � of �, and hence of �, such that � ? �n and � � �gng�1 : But �g�1n � �g�1 � � and � g �1 n ? �n g �1 n : This contradicts to our choice of �n. The following theorem is the main result of this section. Theorem 2.1. Let G be a Polish semigroup [group] and let X be a Polish G-space. Let � and � be measures on X. Denote by r the metric on G and set d is the pseudometric from de�nition 2.2. Then AP (�j�) relative to the metric �(g; h) = maxfd(g; h); r(g; h)g is a separable metric space. A(�j�) [E(�j�) and I(�j�)] is closed in AP (�j�) and complete in this metric. If � = �, then A(�) is a Polish semigroup [E(�) and I(�) are Polish groups]. P r o o f. Clearly that �(g; h) determines a metric on AP (�j�). Let Uk;l;m;q are de�ned in Proposition 2.1 (4). Let Uk;l;m;q be a nonempty set. We can choose a countable set which is dense in Uk;l;m;q in the metric r. Let Q be the union of such sets. Let us show that Q is dense in AP (�j�). Let " > 0; g 2 AP (�j�). It is easily shown that we can �nd Uk;l;m;q such that g 2 Uk;l;m;q and d(g; h) < "; 8h 2 Uk;l;m;q: Let t 2 Q \ Uk;l;m;q such that r(g; h) < ". Then �(g; t) < ". If gn 2 A(�j�)[E(�j�)] and �(gn; g)! 0, then gn tends to g in the initial topol- ogy and fgng is a fundamental sequence in the strong topology. By Lemma 2.2, g 2 A(�j�)[E(�j�)]. Thus these sets are closed. If fgng is a fundamental sequence in the topology which is determined by metric �, then fgng is fundamental in the strong and the initial topologies. Thus fgng tends to an element g 2 G. Then g 2 A(�j�)[E(�j�)] by Lemma 2.3. Hence these sets are complete. The continuity of the group operations follows from Proposition 2.2. Assertions for I(�j�) and I(�) are true since these sets are closed in the strong topology. The theorem is proved. 20 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures R e m a r k 2.2. The condition that G is a group is important. Really. Let X = [0;+1). Set T (x) = � x; x 2 [0; 1) x� 1; x 2 [1;+1) ; S(x) = fxg; where fxg is the fractional part of x. Put � is equivalent to Lebesgue measure on X and let G be the semigroup generated by T and S. Since S2 = S; T kS = S, then G = fST k; T k; k = 0; 1; 2 : : : g. Thus G with the pointwise topology is a Polish semigroup and has one limit point S. Moreover, the strong topology and the initial one are coincide. Evidently that T 2 E(�), S 2 A(�) and T n converges to S strongly. Hence E(�) is not closed. Later on of this section we will consider AP (�j�) with the topology generated by metric �. In particular, the mapping g 7! T�;g(�) and the function g 7! T�;g(�)(E) are continuous, where � 2 L1(�); E 2 B(X). Theorem 2.2. Let G be a subgroup of a Polish group H = X, � and � be measures on H. Then AP (�j�) is a separable metric space with respect to the strong topology. Moreover, A(�j�), E(�j�) and I(�j�) are closed in this topology on AP (�j�). If G is closed, then A(�j�), E(�j�) and I(�j�) are complete. The semigroup A(�) and the groups E(�) and I(�) are Polish. Moreover, the strong topology on A(�j�) is stronger then the topology induced from H. P r o o f. Since I(g; h) = ;, then d(g; h) is a metric. A(�j�), E(�j�) and I(�j�) are closed by Proposition 2.2. Let fgng � A(�j�) (E(�j�), I(�j�)) be a fundamental sequence with respect to the strong topology. Let us prove that fgng is fundamental in the initial topology and hence converges to an element g 2 H. In fact, let x 2 supp �: Then for every neighborhood U of x the following equality is true: lim m;n!1 kT�;gn(�jU )� T�;gm(�jU )k = lim m;n!1 k(�jU )gn � (�jU )gmk = 0: But this is possible i� gn � x is fundamental in H. Hence gn � x converges to an element y 2 H: Thus gn ! yx�1 := g. If G is closed, then g 2 G. By Lemma 2.3, gn converges to g with respect to the strong topology. Therefore A(�j�), E(�j�) and I(�j�) are complete. Now we prove the last assertion. Assume that gn converges to g in the strong topology. By the above, gn converges to some element h with respect to the initial topology. Hence, by Lemma 2.3, gn ! h in the strong topology. Thus g = h and gn ! g with respect to the strong topology. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 21 S.S. Gabriyelyan R e m a r k 2.3. If G is not closed, then, in general, the strong and initial topologies may be not comparable. For example, let G = R;H = T 2. Let p : R ! T 2 be an embedding with the dense image. If � = � = mT2, then the strong topology on T2 coincide with the initial one. Thus the strong topology on R is induced from T 2. Clearly that one is weaker then the initial topology and is not complete. The following proposition shows that AP (�j�) has some local algebraic struc- ture. Proposition 2.3. Let G be a group. For all g 2 AP (�j�) there exists a neighborhood V of g such that h1h �1 2 g 2 AP (�j�); 8h1; h2 2 V: P r o o f. Let g 2 AP (�j�) and � = � + , where �g � �; g ? �: Put a = k�k. Then the set V = fh 2 AP (�j�) : kT�;h(�)�T�;g(�)k < 0; 1a; kT�;h�1(�g)�T�;g�1(�g)k < 0; 1ag is open. Let h1; h2 2 V . Then �g can be represented in the form �g = �1 g + 1g with �1 h �1 2 g � �; 1 h �1 2 g ? � (� = �1 + 1; �1 ? 1): By the choice of V we have k�1 h �1 2 g � �k < 0; 1a end k�1 h �1 2 g k = k�1k � 0; 9a: (2.7) Hence kT �;h1h �1 2 g (�1)k � kT�;g(�1)k � kT �;h1h �1 2 g (�1)� T�;g(�1)k � kT�;g(�1)k � (kT�;h1(�1 h �1 2 g )� T�;g(�1 h �1 2 g )k+ kT�;g(�1 h �1 2 g � �1)k) > 0; 7a (we take into account the following facts: �1 g � ; kTk � 1; �1 h �1 2 g � �; (2:7) and our choice of V ). Thus h1h �1 2 g 2 AP (�j�). Our nearest goal is to prove that the topology generated by product of two measures is the product topology. First, we prove the following lemma. Lemma 2.4. Let nonzero measures �, �1; � � � 2 M+(X) and �, �1; � � � 2 M+(Y ) be norm restricted. Then the following assertions are equivalent: 1. �n � �n ! �� � (by the norm); 22 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures 2. There exist 0 < a � kn � b <1 such that kn�n ! � and �n kn ! �: P r o o f. 1.) 2. Let � and � be normalized convex linear hulls of measures �; �1; : : : and �; �1; : : : , respectively. Let � = F�; �n = Fn�; � = G�; �n = Gn�. By hypothesisZ Y Z X jFn(x)Gn(y)� F (x)G(y)jd�(x)d�(y)! 0; as n!1: (2.8) Let Sn(y) = R X jFn(x)Gn(y) � F (x)G(y)jd�(x). Then they are �-measurable. Choose c > 1 such that �(A) 6= 0, where A = fy : 1=c � G(y) � cg. By the Chebyshev inequality, there exists constant N (not depending on n) such that the following inequality is true: �(Sn � NkSnk) � 1 2 �(A): Thus there exists yn 2 A such that Sn(yn) < NkSnk. Hence, putting kn = Gn(yn)=G(yn), we receiveZ X jFn(x) � kn � F (x)jd�(x) � cNkSnk: Then (2.8) implies Fn(x) � kn ! F (x) or kn�n ! �. Analogously, there exist constants dn such that dn � �n ! �. Therefore �n � �n = (kn�n � dn�n) � 1 kndn ! �� � 6= 0: Hence kndn ! 1. Substituting kn on knp kndn and dn on dnp kndn , we receive a desired sequence (since if knl ! 0 (!1), then � = 0 (� = 0) by the norm boundedness of the sequence �n(�n)). 2.) 1. It is followed from the inequality k�n � �n � �� �k � k(kn�n � �)� �n kn k+ k�� � �n kn � � � k: The lemma is proved. Proposition 2.4. If G = G1�G2;X = X1�X2 and �1��2, then the topology on AP (�) = AP (�1)�AP (�2) is the product topology. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 23 S.S. Gabriyelyan P r o o f. By Proposition 5.1 [7] we have AP (�) = AP (�1)�AP (�2). Let (gn; hn) ! (g; h) in AP (�). Let us show that gn ! g in the topology on AP (�1). Clearly r1(gn; g) ! 0. It is remain to prove that d1(gn; g) ! 0. Let 0 < � 2 L1(�1), 0 < � 2 L1(�2). Then for every (g; h) 2 AP (�) the following equality is true: T�;(g;h)(�� �) = T�((� � �)(g;h)) = T�(�g � �h) = T�1(�g)� T�2(�h): (2.9) By Lemma 2.4, there exist 0 < a � kn � b <1 such that knT�1(�gn)! T�1(�g): Hence it is enough to prove that kn ! 1. By symmetry with respect to � and �, it is enough to prove that limkn � 1. In the converse case, we can assume without loss of generality that kn ! c > 1. Then, by Lemmas 2.4 and (2.9), we have 1 kn T�2(�hn)! T�2(�h) for every part � of �2. Choosing a part � such that �h � �2, we see that k�k k 1 kn T�2(�hn)k � 1 kn k�k ! 1 c k�k; which is a contradiction. By symmetry of the de�nition of the strong topology, it is proved that d1(gn; g)! 0. Hence �1(gn; g)! 0 too. Conversely, let �1(gn; g)! 0 and �2(hn; h)! 0. Clearly that r((gn; hn); (g; h)) ! 0. Suppose 2 L1(�) has the form = �� �, where � 2 L1(�1), � 2 L1(�2). Then (2.9) implies T�;(gn;hn)( ) = T�;(gn;hn)(�� �)! T�;(g;h)(�� �) = T�;(g;h)( ): (2.10) Since every measure 2 L1(�) admits an approximation by �nite sums of measures of the form � � �, (2.10) is true in the general case. Thus �((gn; hn); (g; h)) ! 0. R e m a r k 2.4. For a countable product � = �1 � �2 � : : : the analogical proposition is not true (for example, if � is a right Gaussian measure on R 1). But on �nite products AP (�1) � � � � � AP (�n) (which naturally identi�es with the closed subsets in AP (�) of the forms AP (�1)� � � � �AP (�n)�feg� : : : ) the induced topology from AP (�) is coincide with the product topology. Let us consider some properties of t-ergodic measures. Proposition 2.5. Let (p; �) be a morphism from (G;X) to (H;Y ) and � 2 M(X) t-ergodic. Then �(�) is t-ergodic. 24 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures P r o o f. Let �1 and �2 be pats of �(�). Put � and � are the parts of � such that �(�) = �1; �(�) = �2. Then there exists g 2 G such that �g 6? �. Thus (�1)p(g) = �(�g) 6? �(�) = �2. The constructed topology on AP (�j�) gives an another characterization of t-ergodic measures which explains the word "ergodic". The following proposition is an analog of Proposition 1 [13] (see Proposition 5.5 [9] too). Proposition 2.6. Let G be a group and � 2 M+(X). Then the following properties are equivalent: 1. � is t-ergodic. 2. � is D-ergodic for every countable subgroup D � G such that D \AP (�) is dense in AP (�). 3. There exists a countable subgroup D � G such that � is D-ergodic. P r o o f. 1. ) 2. Let D \AP (�) be dense in AP (�). Let B and B0 be two disjoint D-invariant subsets of positive measure. Then T�;h(1B0�) is concentrated on B0 for all h 2 D. Since D\AP (�) is dense, then T�;h(1B0�) is concentrated on B0 for all h 2 AP (�) too. Hence 1B� ? (1B0�)h for all h 2 AP (�) and therefore for all h 2 G, which is a contradiction. 3. ) 1. Suppose there exist 0 < � � � and 0 < � � � such that � ? �g for every g 2 D. Let Bg be a Borel set such that � is concentrated on Bg and �g(Bg) = 0. Put B0 = \gBg, B = [gg�1B0. Then B is a D-invariant subset of positive measure and �(B) � P g �g(B0) = 0: This contradiction concludes the proof. Proposition 2.7. For a �nite product of groups of transformations a �nite product of t-ergodic measures is t-ergodic. P r o o f. Consider the product of two probability measures. Let Di be count- able subgroups which are generated by countable dense subsets of AP (�i); i = 1; 2. Then �i is Di-ergodic. Let us consider the countable subgroup D = D1�D2. By Proposition 2.4, D \AP (�) is dense in AP (�). By Proposition 2.6, it is enough to prove that � is D-ergodic. Let B be a Borel D-invariant set and �(B) > 0. Let x2 2 X2. Put Bx2 := fx1 : (x1;x2) 2 Bg. Let g1 2 D1. Since (g1; e) �1B = B, then g�1 1 Bx2 = Bx2 . Thus we have either �1(Bx2) = 0 or �1(Bx2) = 1. Set B2 := fx2 : �1(Bx2) = 1g. By the Fubini theorem, we have 0 < �(B) = Z X2 �1(Bx2)d�2 = �2(B2): Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 25 S.S. Gabriyelyan It is remains to show that the set B2 is D2-invariant. Let g2 2 D2. Fixed x2. Since (e; g2) �1B = B, then prX1 B = Bx2 = Bg2�x2 . Thus B2 = g�1 2 B2. The following proposition is an analog of Lemma 1 [5] (the condition of com- mutativity is important). Proposition 2.8. Let G be Abelian. Let � 2 M+(G) and � 2 M+(X) be t-ergodic. Then � � � is t-ergodic too. P r o o f. By Proposition 2.7, � � � is t-ergodic. Put �(g; x) = g � x and p(g; h) = gh. Then (p; �) is a morphism from (G � G;G � X) to (G;X). By Lemma 6.1 [7] and Proposition 2.5, � � � = �(�� �) is t-ergodic too. Notice the following proposition. Proposition 2.9. Let G be a group and � t-ergodic. Let D be a countable subgroup of G such that D \ AP (�) is dense in AP (�). If � 2 M+(X) and D � E(�), then we have either �� � or � ? �. P r o o f. Let E be a Borel set such that j�j(E) > 0 and �(E) = 0. Then, by the hypothesis, we have �g(E) = 0;8g 2 D. Thus the set E0 = [g2Dg�1E is D-invariant and �(E0) = 0. Since � is t-ergodic, then �(E0) = 1. Therefore j�j ? �. 3. Borel type of AP (�j�) The following theorem is the main result of this section. Theorem 3.1. Let (G;X) be a topological semigroup of transformations of a separable metric space X. Let � and � be regular probability measures on X. Then 1. There exists a Borel function �(x; g) : X �G ! [0;1) such that for every �xed G, �(�; g) is a density of the absolutely continuous part of �g relative to �. 2. The sets AP (�j�), A(�j�), E(�j�), I(�j�) and S(�j�) are Borel subsets of G of very bounded types, namely: AP (�j�) is a GÆ�Æ�-set; A(�j�) is a GÆ�Æ-set; E(�j�) is a GÆ�Æ�Æ-set (if G is a group, then E(�j�) is a GÆ�Æ-set); I(�j�) is intersection of a GÆ�Æ-set and a F�Æ�Æ-set; S(�j�) is a F�Æ�Æ-set. 26 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures To prove of Theorem 3.1 we use the de Possel theorem. Let us recall it. Set B(i) to be B if i = 1 and X n B if i = 0. Let fBng be a base of the topology on X. Put N = [nNn; where Nn = n B (i1) 1 \B(i2) 2 \ � � � \B(in) n ; ij = 0; 1; j = 1; : : : ; n o : Denote all distinct nonempty sets from Nn by fAk ng. Then Nn = fAk ng is a �nite family of GÆ-subsets of the separable metric space X. Let � and � be �nite regular measures on X. Then N is a net for �. Consider the next sequence of Borel functions fn(x) = X k:�(Ak n)6=0 �(Ak n) �(Ak n) � �Ak n (x) ; (3.1) D(x) = limn!1fn(x) ; (3.2) where �E is the characteristic function of a set E. De Possel theorem ([15, Ch. IV, �10]) asserts that: if �(x) is a density of the absolutely continuous part of � relative to �, then we have lim n!1 fn(x) = D(x) = �(x) �-a.e. (3.3) Denote by fn(x; g) and D(x; g) the corresponding functions for couple of mea- sures �g and �. Put Im = fk : �(Ak m) 6= 0g, then fm(x; g) = ( �g(A k m ) �(Ak m) ; for (x; g) 2 Ak m �G; if k 2 Im: 0; for (x; g) 2 Ak m �G; if k 62 Im: To prove Theorem 3.1 we need two propositions. Proposition 3.1. Let (G;X) be a topological semigroup of transformations of a separable metric space X. Let � and � be probability measures on X and B(G) a �-algebra of subsets of G. Assume that the following condition is true: (i) there is a net N = S1 n=1Nn = S1 n=1 S1 k=1fAk ng for � in B such that the functions �g(A k n) are Borel for every n; k 2 N: Then 1. There exists a �nite nonnegative B � B(G)-measurable function �(x; g) of two variables such that: 1) For every �xed g the function �(�; g) is a density of the absolutely con- tinuous part of �g relative to �. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 27 S.S. Gabriyelyan 2) For all c 2 R the functions h+c (g) = Z X �+c (x; g)d�(x); h�c (g) = Z X ��c (x; g)d�(x); h(g) = Z X �(x; g)d�(x); are Borel, where �(x; g) � c = �+c (x; g) � ��c (x; g). 3) If E = f(x; g) 2 X �G : �(x; g) > 0g, then the function Q(g) = Z X �E(x; g)d�(x) is Borel. 2. The sets I(�j�); E(�j�); A(�j�); AP (�j�); S(�j�) are Borel. P r o o f. We start with proving Statement 1 of our theorem. 1. If we prove that the function D(x; g) is B�B(G)-Borel and satis�es condi- tions 1�3, then we receive a required function putting �(x; g) = D(x; g) if D(x; g) is �nite and �(x; g) = 1 if D(x; g) is in�nite. First we shall show that D(x; g) is Borel. Since D(x; g) � 0, 8(x; g) 2 X � G, then it is enough to prove that the sets Q = f(x; g) : D(x; g) � cg are Borel for all c > 0. Since the functions �g(A k n) are Borel then the sets L(n; k; "; c) = fg : �g(A k n) > (c� ")�(Ak n)g are Borel for every n; k 2 N and " 2 R (if c is �xed we shall write simple L(n; k; ")). Thus the sets Ak n � L(n; k; ") lie in B � B(G): Put Q" = \1n=1 [1m=n [ImAk m � L(m; k; ") and Q0 = \1p=1Q1=p: (3.4) Clearly that Q0 2 B � B(G): Let us show that Q = Q0. Let (x; g) 2 Q0. Then (x; g) 2 Q1=p, 8p 2 N. Thus for every n 2 N there exist mn � n and kn 2 Imn such that (x; g) 2 Akn mn �L(mn; kn; 1=p), i.e., x 2 Akn mn and g 2 L(mn; kn; 1=p). This is equivalent to the following inequality fmn (x; g) = �g(A kn mn ) �(Akn mn ) > c� 1 p : 28 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures Therefore D(x; g) � c� 1 p . Since p is arbitrary, then D(x; g) � c. Hence Q0 � Q. Conversely, let (x; g) 2 Q. Fixed p 2 N. Then for every n 2 N there exist mn � n and kn 2 N such that x 2 Akn mn and fmn (x; g) > c � 1 p . This equivalent to the following inclusion (x; g) 2 Akn mn � L(mn; kn; 1=p) � [1m=n [Im Ak m � L(m; k; 1=p): Since n is arbitrary, then (x; g) 2 Q1=p. Since p is arbitrary too, then (x; g) 2 \pQ1=p = Q0. Hence D(x; g) is a Borel function on X �G. It been demonstrated above that for a �xed g the function D(x; g) is the density of the absolutely continuous part of �g relative to �. Thus item 1) is proved. Let us prove item 2). Fixed c � 0 and put D(x; g)� c = D + c (x; g)�D � c (x; g). It is enough to prove that the function h+c (g) = R X D + c (x; g)d�(x) is Borel, since, in this case, the functions h(g) = h+0 (g) and h�c (g) = h+c (g) � h(g) + c will be Borel too. Let fn(x; g) � c = f+n (x; g) � f�n (x; g). Since fn(x; g) ! D(x; g), then f+n (x; g)! D + c (x; g) �-a.e. for a �xed g. For � � 0 we put �n(�) = fk : fn(x; g) � c � �g = fk : �g(A k n) � (�+ c)�(Ak n)g \ Ing; where the last equality is true for c + � > 0 (remark that �n(�) depends on g). Then X k2�n(�) �(Ak n) � 1 �+ c X k2�n(�) �g(A k n) � 1 �+ c (a) and this inequality is true for all n 2 N, g 2 G, c+ � > 0. Thus for p > 1 and c+ � > 0, an application of the H�older inequality yields Z fx: p p f + n (x;g)��g p q f+n (x; g)d�(x) = X k2�n(�p) p s �g(Ak n) �(Ak n) � c � �(Ak n) = X k2�n(�p) p q �g(Ak n)� c�(Ak n) � p q (�(Ak n)) p�1 � p s X k2�n(�p) (�g(Ak n)� c�(Ak n)) � p vuuut 0 @ X k2�n(�p) �(Ak n) 1 A p�1 � p s� 1 �p + c �p�1 ; (b) where the last inequality is true since the �rst factor is evidently not larger 1 and for the evaluation of second one we use (a). But for a �xed p > 1 the inequality Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 29 S.S. Gabriyelyan (b) means that the functions p q f+n (x; g) are �-uniformly integrable relative to n (g 2 G is �xed too). Therefore, setting p = (2l + 3)=(2l + 1), we get Hl(g) := Z X p q D + c (x; g)d�(x) = lim n!1 Z X p q f+n (x; g)d�(x) = lim n!1 X k2�n(0) p q �g(Ak n)� c�(Ak n) � p q (�(Ak n)) p�1: (c) Now we consider the Borel functions Rl nk(g; c) = maxf h �g(A k n)� c � �(Ak n) i 2l+1 2l+3 � h �(Ak n) i 2 2l+3 ; 0g: (3.5) Then the sum standing under the limit in (c) isX k Rl nk(g; c) and Hl(g) = lim n!1 X k Rl nk(g; c): (d) Thus the function Hl(g) is Borel. Put (x; g) = q D + c (x; g) +D + c (x; g). Then: 1) (x; g) is Borel �-integrable for every �xed g 2 G (the function q D + c (x; g) 2 L2(�), and by the Cauchy inequality, it is integrable since � is �nite), 2) p q D + c (x; g) � (x; g) for every l 2 N. Since p q D + c (x; g) ! D + c (x; g) as l ! 1, then the Lebesgue theorem yields h+c (g) = Z X D + c (x; g)d�(x) = lim l!1 Z X p q D + c (x; g)d�(x) = lim l!1 Hl(g); 8g 2 G: By (d) this equality can be rewrote in the form h+c (g) = lim l!1 lim n!1 X k Rl nk(g; c): (3.6) Thus the function h+c (g) is Borel and item 2) is proved. Set c = 1. Clearly that D(x; g) > 0, (x; g) 2 E = [1l=1 \1n=1 [1m=nA(m; l); where A(m; l) = f(x; g) : fm(x; g) > 1=lg = [ImAk m � L(m; k; 1 � 1=l) is the union of disjoint rectangles and hence is a Borel set. 30 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures Evidently, �E(x; g) is the limit of nondecreasing Borel functions �El (x; g) where El = \1n=1[1m=nA(m; l). Since 0 � �El (x; g) � 1, then apply the Lebesgue theorem to obtain lim l!1 Z X �El (x; g)d�(x) = Z X �E(x; g)d�(x) for every g. Thus it is enough to show that the functions standing under the limit are Borel. Since �El (x; g) is the limit of nondecreasing sequence of the bounded functions �En l (x; g) where En l = [1m=nA(m; l), then, by Lebesgue theorem, it is enough to prove that the functionZ X �En l (x; g)d�(x); where En l = [1m=nA(m; l); is Borel for all l and n. But A(m; l) = [ImAk m � L(m; k; 1 � 1=l). Therefore En l is a countable union of rectangles. Thus, by the same argument, it is enough to show that the functionR X �A(x; g)d�(x) is Borel, when A is a �nite union of rectangles. But such union can be represented as a �nite union of disjoint rectangles of the form Ai = Qi�Pi, i = 1; : : : ; q; where Qi 2 B, Pi 2 B(G). Hence �A(x; g) = �Q1 (x)�P1(g) + � � � + �Qq (x)�Pq (g). Therefore the functionZ X �A(x; g)d�(x) = �P1(g) Z X �Q1 (x)d�(x) + � � � + �Pq(g) Z X �Qq (x)d�(x) is Borel. This completes the proof. In particular we have Q(g) = Z X �E(x; g)d�(x) = lim l!1 lim n!1 Z X �En l (x; g)d�(x): (3.7) 2. Obviously g 2 AP (�j�) if and only if h(g) > 0. Thus the set AP (�j�) is Borel. Hence S(�j�) = G n AP (�j�) is Borel too. Since g 2 A(�j�) i� h(g) = 1, then A(�j�) is Borel. Clearly E(�j�) = A(�j�) \ fg : Q(g) = 1g. Whence the set E(�j�) is Borel. Evidently, g 2 I(�j�) if and only if �(x; g) = 1; �-a.e. This equality is equiva- lent to the following equalities h+1 (g) = h�1 (g) = 0 (since h+1 (g) > 0 or h�1 (g) > 0 i� �(x; g) > 1 or �(x; g) < 1 on a set of positive measure, but the last is equivalent to the condition g 62 I(�j�)). Therefore the set I(�j�) is Borel. The proposition is proved. In the following proposition the proofs of items 1 and 2 are standard (see [12, Lemma 3] or [3, Prop. 11, Ch. 9, �2, item 6]), though they are the particular cases Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 31 S.S. Gabriyelyan of item 3. Our proof of item 4 is more informative then its group analog (see [12, Lemma 4]). Proposition 3.2. Suppose a topological semigroup G acts continuously on a topological space X with a regular Borel probability measure �. Then 1. If a set U is open, then the function �g(U) = �(g�1U) is lower semicontin- uous. 2. If a set K is closed, then the function �g(K) = �(g�1K) is upper semicon- tinuous. 3. If a set W is open in X �G, then the function R(g) � Z X �W (x; g)d�(x) = �(prX(W \X � fgg)) i s lower semicontinuous. 4. The function �g(E) is Borel for every Borel set E. P r o o f. 3. Let R(g0) = a > 0 and " > 0. Let K be a compact in prX(W\X�fg0g) such that �(K) > a�". SinceW is open, then for every x 2 K there exist neighborhoods U(x) of x and V (x) of g0 such that U(x)� V (x) �W . Since K is compact, then it is covered by some sets U(xi); i = 1; : : : ; n. Set U = [n i=1U(xi) and V = \n i=1V (xi). Then U is a neighborhood of K, V is a neighborhood of g0 and U � V �W . Thus R(g) � �(prX(U � V \X � fgg)) = �(U) � �(K) > a� " for every g 2 V . Note that if we set �(x; g) = g � x;W = ��1(E) and E 2 B, then prX(W \X �fgg) = g�1E. Therefore if we put E = U , then item 1 follows from item 3. 4. It is obvious that this assertion is a corollary of the following trivial lemma (putting L is the family of open sets). This lemma gives a structure of the �- algebra B(L) generated by the family L (a structure of a �-ring see [5, �5, Ex. 9]) and shows that it can be introduced the hierarchy on B(L) which is analogous to the hierarchy of Borel sets. Put !0 = Card N. Lemma 3.1. Let it be done an in�nite family L of subsets of a set X. Suppose the family L is formed by �nite intersections and �nite unions of elements from L. Set A0 = fA n C = A \ (X n C); where A;C 2 Lg. Suppose the family A� consists of all countable unions (intersections) of sets from [�<�A� for every odd (even) ordinal numbers � < !1. Then B(L) = [�<!1A� and CardB(L) � (CardL)!0 : 32 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures R e m a r k 3.1. Since every measure � can be represented in the form � = �+��� where ��; �+ 2M+(X), then the function �g(E) = �+g (E)���g (E) is Borel for all Borel E. P r o o f o f T h å o r e m 2.1. The �rst part follows from Proposition 3.1 immediately. Let us prove the second part. Later on all notations are taken from Proposition 3.1. Moreover, we use the next elementary equalities. If fn(g) converges to f(g) at every point, then fg : f(g) > dg = 1[ c=1 1[ n=1 1\ m=n � g : fm(g) (�) > d+ 1 c � ; fg : f(g) � dg = 1\ c=1 1[ n=c � g : fn(g) > d� 1 c � : Set U r nk forms a decreasing system of open subsets of X such that \rU r nk = Ak n. Put F lr nk(g; c) = maxf h �g(U r nk)� c � �(Ak n) i 2l+1 2l+3 � h �(Ak n) i 2 2l+3 ; 0g: By Proposition 3.2, the function F lr nk (g; c) is lower semicontinuous. Then fF lr nk (g; c)g is a decreasing sequence which converges to Rl nk (g; c). For a �xed n the index k takes a �nite set of values only. Thus the function Z lr n (g; c) :=P k F lr nk (g; c) is correctly de�ned and lower semicontinuous. Moreover, fZ lr n (g; c)g is decreasing by r and converges to the function P k Rl nk (g; c). Then (3.6) implies h+c (g) = lim l!1 lim n!1 lim r!1 Z lr n (g; c): Since h(g)+h�c (g) = h+c (g)+c, then � h(g) > 1 a � � h+1 2a (g) > 1 2a � . Therefore fh(g) > 0g � S1 a=1 � h+1 a (g) > 1 a � . Conversely, if h+1 a (g) > 1 a , then �+1 a (x; g) > 0 on a set of �-positive measure. All the more �(x; g) > 1 a on this set and h(g) > 0. Thus the following equality is true: fh(g) > 0g = 1[ a=1 � h+1 a (g) > 1 a � : By this equality and nonincreasing Z lr n (g; c) with respect to r, we get AP (�j�) = fg : h(g) > 0g = 1[ a=1 � h+1 a (g) > 1 a � Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 33 S.S. Gabriyelyan = 1[ a=1 8< : 1[ b=1 1[ l=1 1\ q=l � g : lim n!1 lim r!1 Zqr n � g; 1 a � > 1 a + 1 b �9= ; = 1[ a=1 8< : 1[ b=1 1[ l=1 1\ q=l " 1[ c=1 1[ n=1 1\ m=n � g : lim r!1 Zqr m � g; 1 a � > 1 a + 1 b + 1 c �#9= ; = 1[ a=1 8< : 1[ b=1 1[ l=1 1\ q=l " 1[ c=1 1[ n=1 1\ m=n " 1\ r=1 � g : Zqr m � g; 1 a � > 1 a + 1 b + 1 c �##9= ; : Hence AP (�j�) is a GÆ�Æ� -set and S(�j�) is a F�Æ�Æ-set. Now we prove the equality 1\ a=1 � g : h(g) > 1� 1 a � = 1\ a=1 � g : h+1 a (g) � 1� 1 2a � : If h(g) > 1 � 1 a , then h+1 a (g) + 1 a > 1 � 1 a and h+1 a (g) > 1 � 1 2a . This proves the inclusion ���. Conversely, let h+1 a (g) � 1� 1 2a and �+1 a (x; g) > 0 on a set E. Then h(g) � Z E �(x; g)d� = Z E � �+1 a (x; g) + 1 a � d� � 1� 1 2a + 1 a �(E) > 1� 1 2a : This proves the inclusion ���. Further, since h(g) � 1 and Z lr m(g; c) is nonincreasing with respect to r, then A(�j�) = fg : h(g) = 1g = 1\ c=1 � g : h(g) > 1� 1 c � = 1\ c=1 � g : h+1 c (g) � 1� 1 2c � = 1\ c=1 ( 1\ a=1 1[ l=a � g : lim n!1 lim r!1 Z lr n � g; 1 c � > 1� 1 2c � 1 a �) = 1\ c=1 ( 1\ a=1 1[ l=a " 1[ b=1 1[ n=1 1\ m=n " 1\ r=1 � g : Z lr m � g; 1 c � > 1� 1 2c � 1 a + 1 b �##) : Thus A(�j�) is a GÆ�Æ-set. Since E(�j�) = A(�j�) \ fg : Q(g) = 1g, it is enough to prove that the set H := fg : Q(g) = 1g is a GÆ�Æ�Æ-set. Since �El is nondecreasing, then by (3.7) and the Lebesgue theorem, we get H = 1\ a=4 1[ l=1 � g : lim n!1 Z X �En l (x; g)d�(x) > 1� 1 a � 34 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures = 1\ a=3 1[ l=1 " 1\ n=1 � g : Z X �En l (x; g)d�(x) > 1� 1 a �# ; (since �En l (x; g) is nonincreasing) where (recall that c = 1) En l = [1m=nA(m; l) = [1m=n [k2Im Ak m � L(m; k; 1� 1=l): (3.8) Since Ak m = \rU r nk and L(m; k; 1 � 1=l) = � g : �g(A k m) > 1 l �(Ak m) � = 1[ t=1 1\ r=1 � g : �g(U r mk) > � 1 l + 1 t � �(Ak m) � ; then Ak m � L(m; k; 1 � 1=l) = 1[ t=1 1\ r=1 � U r mk � � g : �g(U r mk) > � 1 l + 1 t � �(Ak m) �� : Denote the set standing in the round brackets by V rt mk . By Proposition 3.2, this set is open. Taking into account (3.8), we get En l = 1[ m=n [ k2Im 1[ t=1 " 1\ r=1 V rt mk # = 1[ q=1 E nq l ; where E nq l = S n+q m=n S k2Im S q t=1 �T1 r=1 V rt mk � � form an increasing sequence of GÆ- sets. Let E nq l = \1 s=1W nq sl , where W nq sl is a decreasing sequence of open subsets of X �G. Then �En l (x; g) = lim q!1 lim s!1 �Wnq sl (x; g): Taking into account of the character of convergence and the Lebesgue theorem, we get H = 1\ a=2 1[ l=1 2 4 1\ n=1 2 4 1[ q=1 2 4 1\ s=1 8< :g : Z X �Wnq sl (x; g)d�(x) > 1� 1 a 9= ; 3 5 3 5 3 5 : By Proposition 3.2, the set standing in the �gure brackets is open. Thus H is a set of the type GÆ�Æ�Æ . If G is a group, then E(�j�) = A(�j�)\[A(�j�)]�1 [7, Theorem 1.2]. Therefore E(�j�) is a GÆ�Æ-set. Clearly that g 2 I(�j�), �(x; g) = 1 � � a.e. , g 2 � g : h�1 (g) = 0 \� g : h+1 (g) = 0 : Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 35 S.S. Gabriyelyan If h+1 (g) = 0, then h�1 (g) = h+1 (g)�h(g) + 1 equal to zero if and only if h(g) = 1. Hence I(�j�) = A(�j�) \ fg : h+1 (g) = 0g. Since fg : h+1 (g) > 0g is a GÆ�Æ�-set (see the proof for AP (�j�)), then I(�j�) is the intersection of the GÆ�Æ-set and the F�Æ�Æ -set. Theorem is proved. R e m a r k 3.2. If G is a separable metric semigroup, then �(x; g) is a function of the GÆ�Æ�-type, i.e., the inverse image of an open set is the GÆ�Æ�-set. Really. Let U r nk be open sets in X such that \rU r nk = Ak n. Then L(n; k; ") = [1b=1 \1r=1 � g : �g(U r nk) > � c� "+ 1 b � �(Ak n) � : By Proposition 3.2, this set is a GÆ�-set. Thus A k m � L(m; k; ") is a GÆ�-set and its complement is a F�Æ-set. Formula (3.4) follows f(x; g) : D(x; g) > cg = [1q=1 \1p=1 \1n=1 [1m=n [ImAk m � L � m; k; 1 p ; c+ 1 q � ; f(x; g) : D(x; g) < cg = [1p=1[1n=1\1m=n\Im (X�G)n � Ak m � L � m; k; 1 p ; c �� ; f(x; g) : D(x; g) =1g = \1c=1 \1p=1 \1n=1 [1m=n [ImAk m � L � m; k; 1 p ; c � : Thus, taking into account that G is separable and metric, all these sets and their intersections are GÆ�Æ�-sets. Let U be open. Then U = [a(ca1; ca2), where ca2 � ca+1 1 . Hence (if 1 2 U) ��1(U) = [af(x; g) : ca1 < D(x; g) < ca2g ([f(x; g) : D(x; g) =1g) is a GÆ�Æ�-set. R e m a r k 3.3. Let G be a separable metric group, � = � and I = f(x; g) : g � x = xg. Then we may assume that �(x; g) satis�es the following conditions: 1. �(x; g) is the function of at most GÆ�Æ�Æ-type. 2. �(x; g) = 1; 8(x; g) 2 I: 3. �(x; g) > 0; 8g 2 E(�): In particular, if G = E(�), then ln�(x; g�1) is at most GÆ�Æ�Æ-type cocycle. Really. Since I is closed in X � G, then Ig = fx : (x; g) 2 Ig 2 B(X). Evidently, for all E � Ig we have g�1E = E. Thus �gjIg = �jIg . Hence �(x; g) = 1; �-a.e. on Ig. By Remark 3.2 and Theorem 3.1, the set A = ��1(0) \X �E(�) 36 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures is a F�Æ�Æ -set. Put ~�(x; g) = �(x; g) for (x; g) 62 I [A; and ~�(x; g) = 1; for (x; g) 2 I [A: Thus, if 1 62 U (if 1 2 U), then ~��1(U) = ��1(U) f(X �G) n (I [A)g � ~��1(U) = ��1(U) [ I [A � is a GÆ�Æ�Æ -set. The cocycle inequality follows from proposition 3.2 [7]. For the set I(�j�) Theorem 3.1 may be improved. Proposition 3.3. Let G and X be separable metric spaces. Then the set I(�j�) is closed in G. P r o o f. By Proposition 3.2, the function �g(K) is upper semicontinuous. Thus, if gn tends to g, then limn!1�gn(K) � �g(K): Therefore, by Theorem 2.1 [2], for every bounded real continuous function f(x) (and since �gn = c�) we get c Z fd� = Z fd�gn ! Z fd�g: Since every measure is determined by its values on such functions completely, then �g = c� and g 2 I(�j�). Moreover, we can repeat the proof of Theorem 1.2.4 [8] word for word and prove the following proposition. Proposition 3.4. If X = G is a separable metric group, then the subgroups Il(�), Ir(�), It(�) are compact. R e m a r k 3.4. In the general case this proposition is not true. In fact, let G = R;H = T 2 and p : R ! T 2 be an embedding with the dense image. Evidently, if � = mT2, then Il(�) = R. R e m a r k 3.5. Note that, if X = G, then the representation of E(�) in U(L2(�)) determined by the equality Sg (f) (x) = s d�g d� (x)f � g�1 � x � = p �(x; g)f � g�1 � x � ; is exact. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 37 S.S. Gabriyelyan Let G be a group, then the group E(�) plays special role among considerable sets. It is naturally to raise the question of description such groups. According to Theorem 3.1, E(�) is a set of very bounded type. Now we give some open questions. 1. For which subgroups H � G there exists � 2M+(G) such that H = E(�)? 2. What is the Borel class of E(�) exactly? For example, let ! be a right Gauss measure on R1 . Then E(!) = l2 [16, � 5, Th. 1]. Let us prove that l2 is a F� nGÆ-set. Really, let fn(x) = x21 + � � � + x2n;x = (x1; : : : ; xn; : : : ) 2 R1 , be continuous functions on R1 . Since l2 = 1[ k=1 1\ n=1 fx : fn(x) � kg ; then l2 is a F�-set. In the other side, l2 is not GÆ-set by the results of [10, Ch. VI, � 34]. 3. What are the classes for which there exists a group of quasiinvariance of a probability measure? For Polish groups the Mackey�Weil theorem [11] may be formulated the fol- lowing way. Theorem. Let X = G be a Polish group. Then the following propositions are equivalent: 1. There exists a measure � such that E(�) is open. 2. G is local compact. All examples of E(�) which are known to the author are sets of F�-type. 4. How are connected properties of G with Borel classes of all E(�)? Proposition 3.5. Let X = G be a Polish group. Then the following proposi- tions are equivalent 1. E(�) is a set of GÆ-type for all probability measures �. 2. G is discrete. P r o o f. Obviously, it is enough to prove the su�ciency. Let H be a coun- table dense subgroup of G. Set � = P h2H �hÆh, P h2H �h = 1, �h > 0. Then E(�) = H. By [10, � 34, Th. 3 and � 9, Th. 4], H is a GÆ-set only if G = H. Naturally, this questions are considered for the most important cases when G is either local compact or Abelian or X = G. 38 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 1 Topological Properties of the Set of Admissible Transformations of Measures References [1] J. Aaronson and M. Nadkarni, L 1 eigenvalues and L2 spectra of nonsingular trans- formations. � Proc. London Math. Soc. 55 (1987), 538�570. [2] P. Billingsley, Convergence of probability measures. 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