REAL ANALYSIS, XXIV:
ABSTRACTS
Abstract: We study the notions of absolute continuity of Rado, Reichelderfer and Malý.
For a fixed symmetric convex set of non-empty interior let denote the set of all `balls' of Rn in the norm defined by K0, that is, we put .
A function is called absolutely continuous with respect to , or f is -AC, if for every positive there exists a positive , such that for every disjoint system of sets from . Analogously, f is of bounded variation with respect to , or f is -BV, if where the supremum is taken over every sequence of disjoint sets . We say that f satisfies the -Rado-Reichelderfer condition, or f is -RR, if there exists an absolutely continuous finite measure on Rn, for which holds for every . We say that f satisfies the weak--Rado-Reichelderfer condition, or f is -RR*, if there exists a finite (not necessarily absolutely continuous) measure , for which holds for every . We prove that -RR -AC -BV -RR*, and all these notions depend on .
Professor Zoltan Buczolich, Main Lecture, Session 1
Birkhoff Sums and zero-one Laws
Abstract:
In this talk we focus on one direction of progress being made since our last invited talk at the XXI'st Summer Symposium on Real Analysis. In the abstract [B1] we claimed that we solved a problem of M. Laczkovich concerning whether there are two irrational rotations and of the unit circle such that the Birkhoff- (ergodic) averages of a measurable function converge almost everywhere to 0, while the averages converge almost everywhere to 1. During the refereeing process of that result it was suggested to state and prove a more general result [B2]. We discuss this in the first half of the talk. In Hungary, at the 1998 Miskolc Conference on Dimensions and Dynamics we talked about this result. R. D. Mauldin suggested to look at an old unsolved problem of J. A. Haight and H. von Weizsäcker. This problem asked whether it is true that for an arbitrary measurable function defined on the positive reals either either converges almost everywhere or diverges, that is, whether there is a zero-one law for this function. After the conference we solved this problem and published in [BM]. In fact, we found a counterexample function showing that there is no zero-one law. Later together with J-P. Kahane further progress was being made with respect to the additive generalization of this problem [BKM1] and [BKM2]. We discuss this generalized problem in the end of the talk. Our results prompted some further research and some of these results, as far as I know, will be discussed in other talks during this meeting.
[B1] Z. Buczolich, Generalized Integrals and Related Topics, Real Anal. Exchange, Vol. 23 (1), 1997/8 pp. 27-36.
[B2] Z. Buczolich, Ergodic averages and free Z2 actions, Fund. Math. 160 (1999) no.3, 247-254.
[BM] Z. Buczolich and R. D. Mauldin, On the convergence of for measurable functions, Mathematika (to appear).
[BKM1] Z. Buczolich, J-P. Kahane and R. D. Mauldin, Sur les séries de translatées de fonctions positives, C. R. Acad. Sci. Paris Sér. I Math. 329, no. 4, 261-264.
[BKM2] Z. Buczolich, J-P. Kahane and R. D. Mauldin, On series of translates of positive functions, (submitted).
Professor Gregory Sokhadze, Lecture 2, Session 1
On equivalence of distributions of solutions of stochastic differential equations
Abstract: For the distributions of solutions of a boundary problem corresponding to an ordinary stochastic differential equation, the conditions of their equivalence are given and the Radon-Nikodym derivatives are calculated.
Professor Robert Vallin, Lecture 3, Session 1 Cantor Sets Thickness, and Porosity
Abstract: Porosity has been used for many years to describe small sets. Thickness is a recently developed indicator of how large a Cantor set is. In this talk w e realte the thicknesss of a Cantor set to both the usual notion of porosity and Denjoy's porosity index. We then apply this to the sum of Cantos sets and give some open questions.
Professor Chris Ciesielski, Lecture 4, Session 1
On anti-Schwartz functions and functions with two element range
Abstract: Krzysztof Ciesielski, Kandasamy Muthuvel, and Andrzej Nowik
A function is antisymmetric (anti-Schwartz) at provided there is no sequence such that f(x+hn)=f(x-hn) ( f(x+hn)=f(x-hn)=f(x), respectively) for every n. We will characterize the sets S(f) of all points of antisymmetry of f and show that f cannot be anti-Schwartz at all points.
It is also shown that if at each point x we ignore some countable set from which we can chose sequence hn, then we can construct a function which at every point is anti-Schwartz in this sense if and only if the continuum hypothesis holds.
Dr. Marta Babilonova, Lecture 5, Session 1
Solution of a problem of S. Marcus concerning J-convex functions
Abstract: This lecture will be devoted to a characterization of the class S of stationary sets for J-convex functions , where is a convex oprn subset of Rn (this problem was proposed by Prof. S. Marcus on the last Summer Symposium in Poland).
Professor Peter Wingren, Lecture 6, Session 1
Smooth Functions Defined on Fractal Sets
Abstract: This talk is about fractals and functions defined on fractals. Three matters are discussed. 1) How do we know if a function is smooth if we just know its values on a (fractal) subset of the Euclidean space? 2) How are smoothness on fractals and smoothness on the whole Euclidean space related? 3) A new characterization is given of those function which are smooth in the sence that they are bounded and their second difference are bounded by a constant times the step length. This characterization is possible to use to connect smoothness on fractals and smoothness on Euclidean space. A new extension operator is presented.
Professor Krzysztof Plotka, Lecture 7, Session 1
Sum of Sierpinski-Zygmunt and almost continuous functions
Abstract: It is shown that the statement: every function from R into R can be represented as a sum of Sierpinski-Zygmunt and Almost Continuous functions, is independent of ZFC axioms. It is also proved (in ZFC) that not every function from Rn (n>1) into R has such representation.
SESSION 2, WEDNESDAY AFTERNOON
Professor Slava V. Chistyakov, Lecture 08, Session 2
On mappings of finite generalized variation and nonlinear operators
Abstract: The talk is concerned with mappings of bounded variation defined on an interval (or a subset) of the reals and taking values in a normed, metric or uniform space. The notions of variation are considered in the senses of Jordan, Riesz, Wiener, Young and Orlicz. The presentation is organized in such a way that the results for different types of variation (although different in proof) look almost the same. The principal topics are structural theorems and Helly type selection principles. We show that compact-valued set-valued mappings of bounded generalized variation admit regular selections of bounded generalized variation (in the sense considered). More general problem is the existence of regular solutions of bounded generalized variation to the functional inclusion of the form , , with the multivalued right hand side F. In this respect we study nonlinear multivalued Nemytskii type superposition operators between different spaces of mappings of generalized variation and present a complete characterization to Lipschitzian operators of this type. Finally, we present possible extensions of the above results for mappings of two (or more) variables of finite variation in the sense of Hardy whose values lie in metric or uniform spaces and discuss set-valued mappings of this type.
Professor Kenneth Falconer, Main Lecture, Session 2
Nonlinear Mercerian Theorems
Abstract: This talk covers a circle of ideas concerning the existence of limits of real valued functions. Variants of the classical theorem of Mercer will be related to dynamical systems, to finding dimensions of fractals, to number theory and to serving drinks at parties.
Professor Maarit Jarvenpaa, Lecture 09, Session 2
Porosities of measures
Abstract: We prove that in the real line packing dimension of any Radon measure isbounded above by a function depending on porosity. This upper bound tends tozero as porosity tends to its maximum value.
Professor Henry Fast, Lecture 10, Session 2
All translates of perfect in all directions sets meet Cantor-like products.
Abstract: For a subset P of whose projections on all the factors of are perfect linear sets and a number there is a based on product D of Cantor-like linear sets with the property that the intersection of every translate of P with D is -dense in P. A version of this result shows that for a countable class of of perfect in all direction subsets of there is a based on product D* of 'condensed' Cantor-like linear sets such that intersection of every translate of a Pi with D* is dense in Pk.
Professor Dave Lawrence Renfro, Lecture 11, Session 2
Mean Porous Sets
Abstract: Mean porosity was introduced in a paper by Pekka Koskela and Steffen Rohde, where it was stated ... it is not hard to giveexamples of mean porous sets that fail to be porous.** Meanporosity was further applied in a paper by Feliks Przytyckiand Steffen Rohde
Marton Elekes, Lecture 12, Session 2
Linearly ordered sets of real functions
Abstract:
Any class of real valued functions is partially ordered by the natural pointwise ordering. Our aim is to characterize the possible order types of the linearly ordered subsets of . For some classes, the problem is easy, for others it is open and sometimes it turns out to be independent of ZFC.
Professor Paolo de Lucia, Lecture 13, Session 2
Liapunoff theorems in noncommutative measure theory
Abstract:
Let G be a commutative Hausdorff topological group. Let m be a G-valued, completely additive measure on a complete orthomodular poset, L. It is shown, among other results, that when the centre of L is non-atomic, then m must be strictly bounded. When L is specialized to being the lattice of projections in a von Neumann algebra this extends some results known for real valued measures.
Mr. Xianfu (Shawn) Wang, Lecture 14, Session 2
Typical subdifferentiability of continuous functions on separable Banach spaces
Abstract: We prove a ``typical subdifferentiability principle and apply it to a variety of complete metric spaces of continuous functions on separable Banach spaces; so as to obtain existence offunctions with maximal subdifferentials when ordered by inclusion. The relationship between continuous functions with maximal subdifferentials and nowhere monotone functions is discussed.
Professor Kiko Kawamura, Lecture 15, Session 2
Computational Complexity of Self-similar sets
Abstract: The aim of this study is to find a mathematical tool other thanfractal dimensions and considered to be an estimation of complexity in fractal geometry. We investigate self- similar sets from viewpoint of Pour-El and Richards style computable analysis, and propose computational complexity as one of the tools to estimates complexity of self-similar sets.
Professor Maria Eugenia Mera Rivas, Lecture 16, Session 2
A Zero-One Half Law for Porosity of Measures
Abstract: We introduce two definitions of upper porosity of a measure which range from 0 to 1/2 and from 0 to 1 respectively, and prove that actually the first porosity only can take the extreme values 0 or 1/2, and the second one takes either the value 0 or the values 1/2 or 1. We also prove that any measure m which does not satisfy the doubling condition m-a.e has a maximal porosity.
Dr. Roy Mimna, Lecture 17, Session 2
Persistence stability of non-invertible maps on compact metric spaces
Abstract: Let C(X,X) denote the set of all continuous non-invertible maps from X into X, w here X is either an arbitrary compact metric space, or a connected and locally c onnected compact metric space with the fixed-point property. We show that there are open dense subsets of C(X,X) which possess persistence-stability properties with respect to attractors and basins of attraction. Milnor attractors are dis cussed, and results are presented on strongly chaotic functions and stability.
Professor Francis Jordan, Lecture 18, Session 2
Blumberg theorem for sets of functions
Abstract: Suppose you have a collection of functions from R into R of a certain type say Baire Class 1. Can you find a dense set upon which all of these functions are continuous h ow about a perfect set? We investigate this and related questions.
SESSION 3, THURSDAY MORNING
Professor Siegfried Graf, Lecture 19, Session 3
Asymptotics in quantization of random vectors
Abstract: Given a natural number n and a random variable X taking values in d-dimensional Euclidean space the quantization problem is to find a best approximation to X by a random variable Y which attains at most n different values in the same space. The distance is measured by the usual r-norm. Let e(n,r) be the error made by such an approximation. The asymptotic behaviour of e(n,r) for n tending to infinity is studied and related to several types of fractal dimensions for measures.
Professor Yuval Peres, Main Lecture, Session 3
When Projections Reduce Dimension What is Preserved?
Abstract: We learn in physics that when a system seems to lose energy (e.g. a sliding object slowing down due to friction) the lost energy is really converted to another form (often, heat) and overall energy, PROPERLY DEFINED, is preserved.
In this talk we describe an analogous principle for (generalized) projections of measures, discovered in joint work with Wilhelm Schlag: When a typical projection-type map P reduces the (correlation) dimension of a measure , a (typically) preserved quantity is the SOBOLEV DIMENSION, which accounts both for the correlation dimension of a measure, and for the degree of smoothness of its density (if it exists). Thus, here smoothness plays the role of heat. A notion of transversality for the ensemble of projection maps is needed. Some sample applications:
I will briefly survey these applications, indicate the meaning of transversality in each setting, and state the key open problem.
Artemi Berlinkov, Lecture 20, Session 3
On packing measure and dimensions of random fractals
Abstract: We consider fractals generated by a finite system of similarities with random coefficients (T1,...,Tn). It has been shown that its Hausdorff dimension equals , so that and most of the time the - Hausdorff measure of the fractal is 0. It turns out that the packing and box- counting dimension are also and in the same cases the -packing measure is infinite.
Professor Kandasamy Muthuvel, Lecture 21, Session 3
Darboux fuction and continuous function
Abstract: It is known that if gof is continuous, and f is both continuous and surjective, then g is continuous (Real Anal. Exch. 18, No 2 (1992-93), 420-426). We generalize the above result by showing that if gof is continuous, and f is both Darboux and surjective, then g is continuous. This also improves Theorem 3 in Real Anal. Exch. 23, No 1 (1997-98), 211-216. We also prove that continuous and Darboux can be interchanged in the above statement of our result, i.e., if gof is Darboux and f is both continuous and surjective, then g is Darboux.
Professor Ondrej Zindulka, Lecture 22, Session 3
A set that is fat and slim at the same time
Abstract: Under a mild set-theoretic assumption, there is a set in the plane that is of universal measure zero and yet it meets every nontrivial curve. In particular, it is of positive topological dimension. I a model of ZFC such a set fails to exist. The idea is exploited in order to get similar results involving universal meager sets and other small sets in analytic metric spaces.
Professor Svetlana V. Butler, Lecture 23, Session 3
Approximation in the space of quasi-measures
Abstract: A quasi-measure is a set function on a compact Hausdorff space which is monotone, regular, additive, but not necessarily subadditive. Quasi-measures correspond to functionals that are linear on singly-generated subalgebras. We will consider various approximations in the space of quasi-measures involving classes of simple, representable, extreme quasi-measures and ordinary measures.
Professor Michal Rams, Lecture 24, Session 3
Packing dimension estimate for the exceptinal parameter set
Abstract: The Hausdorff dimension of the limit set of iterated function system with overlaps must not be greater than so called similarity dimension of the ifs. It may, however, be strictly smaller. Given a family of ifs, the parameters for which it happens are called exceptional. Assuming some transversality conditions on the family, we estimate from above the packing dimension of the exceptional parameters set.
Professor Matteo Rocca, Lecture 25, Session 3
Smoothness conditions and differentiability properties of real functions
Abstract: Authors: Davide La Torre and Matteo Rocca
The aim of this talk is to present a survey of known results and some new ones connecting smoothness-type and differentiability properties of real functions. A boundedness requirement of second-order (symmetric) divided differences is called (loosely) a smoothness condition (see Thomson, 1994). This kind of condition is closely related to continuity and differentiability (in the usual or in the Peano sense) of real functions (see for instance Thomson, 1994, Marcinkievicz and Zygmund, 1936). A requirement stronger that smoothness is the existence of the so-called Riemann-Schwarz derivatives. Most (but not all) of these relations hold also when one-sided differences are considered. In this talk, in the first place we show generalizations to orders higher than two of some results linking uniform smoothness and differentiability properties (see e.g. Thomson, 1994). Most of considered results are local in the sense that they connect the smoothness or the existence of Riemann derivatives on an interval (or on measurable set) with local continuity and differentiability properties. Driven by this fact, we give results linking boundedness of a certain type of divided differences and differentiability properties of a function at a point. Furthermore we show the equivalence between the existence of Riemann-type derivatives and Peano derivatives at a given point.
References
Marcinkievicz J., Zygmund A., On the differentiability of functions and summability of trigonometric series. Fund. Math., 26, 1936, 1-43.
Thomson B.S., Symmetric properties of real functions. Marcel Dekker, New York, 1994.
SESSION 4, THURSDAY AFTERNOON
Professor Ori Sargsyan, Lecture 26, Session 4
On the convergence and Gibbs phenomenon of Franklin series
Abstract: We present theorems on convergence and uniform convergence at a point for simple Franklin series,as well as convergence and uniform convergence at a point for double franklin series . The results are applied to the study of Gibbs phenomenon of Franklin system. And also is obtained , that the Franklin series will be Fourier- Franklin series of function of the bounded variation if and only if the variations of the partial sums will be uniform bounded .
Professor Martina Zaehle, Lecture 27, Session 4
Riesz potentials of fractal measures
Abstract: Riesz potentials of fractional order of Ahlfors regular Borel measures with compact support in euclidean space are investigated. They define compact self-adjoint operators on the space of square integrable functions. The spectral dimension of an associated Laplace operator agrees with the Hausdorff dimension of the (fractal) support. For the marginal case of Lebesgue measure we obtain the classical boundary free euclidean Laplacian.
Professor Jean-Pierre Kahane, Main Lecture, Session 4
Baire category theorem and trigonometric series
Abstract: Professor Kahane is being honored at the University of Uppsala during this week and will be unable to deliver his lecture in person. With the help of Francois Guenard, though, he has produced a video recording of his lecture which will be presented at the scheduled time in ENV 110. Questions following the lecture will be more problematic, however.
Professor Oleg Kovrijkine, Lecture 28, Session 4
On periodizations of functions in higher dimensions
Abstract: We prove that the norm of periodizations of a function from is equivalent to the norm of the function itself in higher dimensions. We generalize the statement for functions from where in the spirit of the Stein- Tomas theorem.
Professor Giuseppa Riccobono , Lecture 29, Session 4
A note on the PU-integral on an abstract space
Abstract: We consider a PU-integral (i.e. an integral defined by partition of the unity)on a topological abstract measure space and compare it with the Lebesgue integral.
Professor Michal Morayne, Lecture 30, Session 4
Martingales and Strong Differentiation of Integrals
Abstract: A martingale proof of Jessen-Marcinkiewicz-Zygmund theorem on strong differentiation of integrals is outlined. The proof uses Cairoli's multiindex martingales convergence theorem.
Professor Ralph E. Svetic, Lecture 31, Session 4
Fixed points and the Composition of Darboux Baire One Functions
Abstract: We describe several results related to a question of Ciesielski: Has the composition of two derivatives (DB1 or Ext) functions from I to I a fixed point?
Professor Stanley C. Williams, Lecture 32, Session 4
Positivity of disintegration kernals of random measures generated by cascading exchangeable processes.
Abstract: Cascading liklyhood ratios associated with exchangeable processes leads to random measures which are generalizations of martingales of Mandlebrot, and which fall within the theory of T- martingales of Kahane. Disintegrations of these measures in terms of ergotic limits are investigated in terms of linkage and separation. The effects of X-factors leading to multiplicative cascades are studied.
Professor Grigore Ciurea, Lecture 33, Session 4
On the Henstok and McShane integrability
Abstract: It is known that any integrable function is McShane integrable, and in fact we only McShane's proof in ``Stochastic calculus and stochastic models'' Academic Pr ess, New York (1974). In the first part of this paper we provide another proof o f this result, and in the second part, we shall study the Lebesgue mensurability of the Henstok integrable functions. Thus, we shall show that any function, whi ch is Henstok integrable, is Lebesgue measurable, and thus we shall prove in a d ifferent way from the former proof from Y. Kubota in [3] the Lebesgue mensurabil ity of the McShane integrable functions. We also present a few corollaries of the others two result, some of them being known already.
SESSION 5, FRIDAY MORNING
Professor Jaroslav Smital, Lecture 34, Session 5
Minimal sets of continuous maps of the interval and distributional chaos.
Abstract: Restrictions of continuous maps of the interval to their minimal sets give interesting information on the behavior of continuous mappings on compact metric spaces. Phenomena, that are not possible for continuous maps on the whole interval, can there appear. We look for relations with distributional chaos.
Professor Michal Misiurewicz, Main Lecture, Session 5
Rotation Theory
Abstract: In the theory of discrete dynamical systems one looks at the iterates of a single map of a phase space into itself. We can say that we know the system if we know the behavior of orbits of points from the phase space. Thus, we have to develop tools that allow us to draw conclusions about large sets of orbits from a limited initial information on the system. One such tool is Rotation Theory.
Rotation numbers were defined first by Poincare for orientation preserving homeomorphisms of a circle. They measure the average angle by which the points are moved along a circle. For all orbits we get the same rotation number, and this number is the main characteristic of the homeomorphisms. This notion has been later generalized to the cases of orientation preserving homeomorphisms of an annulus, degree one maps of a circle, and homeomorphisms of a torus isotopic to the identity. In those cases instead of one rotation number we get sets of rotation numbers (rotation sets). Often existence of two or three orbits with given rotation numbers (rotation vectors for the torus case) implies existence of uncountably many orbits with different behaviors. If the orbits that we know are periodic then the initial information is finite, yet we can draw conclusions on chaoticity of the system.
In the examples given above, to get the rotation numbers or vectors, we are computing the average displacement along the trajectories. However, Rotation Theory has been generalized to the situations where one uses functions other than the displacement. This gives especially good results in the investigation of periodic orbits of interval maps.
Professor Grazyna Kwiecinska, Lecture 35, Session 6
A theorem about Caratheodory's superposition of multivalued maps
Abstract: A new concept of derivative of multivalued map is introduced. Let X be a measure space, Y a Polish one and Z a Banach space. Let be a multivalued map having equicontinuous x-sections and y-sections being derivatives. Then for every continuous function Caratheodory's superposition G(x)=F(x,f(x)) is a derivative. Some application of this theorem to the differential equations with multivalued right-side is shown.
Professor Luisa Di Piazza , Lecture 36, Session 5
A characterization of variationally McShane integrable Banach-space valued functions
Abstract: We present a complete characterization of the variationally McShane integral for Banach-space valued functions defined an - finite,outer regular quasi-Radon measure space. As corollary of this characterization we get a generalization of a W. Congxin- Y. Xiabo's result for Banach-space valued functions defined on closed interval endowed with the Lebesgue measure. Moreover, using this characterization we also generalize a result of V. Skvortsov and V. Solodov which proved that the McShane integral and the variationally McShane integral are equivalent, when considered on a closed interval equipped with the Lebesgue measure, if and only if the range space isof finite dimension.
Professor Tomasz Natkaniec, Lecture 37, Session 5
Universally Kuratowski-Ulam spaces
Abstract: The talk will be based on the paper ``Universally Kuratowski-Ulam spaces`` by David Fremlin, Tomasz Natkaniec and Ireneusz Reclaw.
A pair (X,Y) of topological spaces is called Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds in . A space Y is called universally Kuratowski-Ulam space if (X,Y) is a Kuratowski-Ulam pair for every space X.
It is known that every space with countable -basis is uK-U. We prove that there are uK-U Baire spaces wich do not have countable -basis, but every Baire uK-U space is ccc.
We consider also subspaces, products, unions and continuous images of uK-U spaces.
Professor Lech Bartlomiejczyk, Lecture 38, Session 5
Solutions with big graph of iterative functional equations
Abstract: We consider several types of iterative functional equations (e.g. general equation of the first order, equation of invariant curves, equation of iterative roots) and we look for its solutions which have a big graph. The graph of such a solution has some strange properties: is topologically big, has full outer measure; but in some cases it is also dense.
Professor Eduard Belinsky, Lecture 39, Session 5
Metric entropy of subsets of absolutely convergent Fourier series
Abstract: We estimate the metric entropy of compact subsets of Ap for 0<p<2. The interesting phenomena is found: for 1<p< 2 the entropy depends on the smoothness but for the entropy is independent of the smoothness.
Professor Jolanta Wesolowska, Lecture 40, Session 5
Investigation of sets of convergence points of sequences of some real functions
Abstract: A subset A of a Polish space X is of type iff there exists a sequence of continuous real functions defined on X convergent exactly at each point x of A (Sierpinski, Hahn). The problem we deal with is to find the analogous characterization for sequences of functions from other classes. Moreover, the sets of point of convergence to infinity are examined.
Dr. Bill Beyer, Lecture 41, Session 5
History of the ham sandwich theorem: Steinhaus to the internet
Abstract:
SESSION 6, SATURDAY MORNING
Professor Alexander Kharazishvili , Lecture 42, Session 6
On measurability properties of subgroups of a given group
Abstract: Let denote the classical Lebesgue measure on the real line R. It is well known that is invariant under the group of all motions of R and, moreover, there are invariant (under the same group) measures on R strictly extending (see, e.g., [2], [3], [6]). The following problem arises naturally: give a characterization of all those sets , for which there exists at least one invariant measure on R extending and satisfying the relation . An analogous problem can be posed for subsets of R measurable with respect to various quasi-invariant extensions of . These two problems essentially differ from each other and none of them is solved at the present time.
The class of all subgroups of R may be regarded as a class of subsets of R, which distinguishes these problems. More precisely, for any group , there exists a quasi-invariant extension of such that ; at the same time, there exists a subgroup H of R such that, for each invariant extension of , we have (notice that H can easily be constructed by using a Hamel basis of R).
The following statement yields a more general result. We recall that a topological group is standard if coincides with some Borel subgroup of a Polish group.
Theorem 1. Let be a standard group equipped with a Borel (left) quasi-invariant probability measure and let G1,G2,...,Gn be an arbitrary finite family of subgroups of . Then there exists a (left) quasi-invariant extension of such that .
The proof is based on the fundamental Mackey theorem [5] and some
auxiliary results presented in [3]. In this connection,
it should be noted that if
H1,H2,...,Hk are any subgroups of satisfying the
relations
Dealing with countable families of subgroups of , we come to a significantly different situation. For example, it is not hard to show that there exists a countable family of subgroups of R, such that cannot be extended to a quasi-invariant measure whose domain includes all these subgroups. The next result generalizes the above-mentioned fact.
Theorem 2. Let be an uncountable divisible commutative group. Then there exists a countable family of subgroups of , such that:
1) for each , we have ;
2) .
In particular, for any probability quasi-invariant measure on , at least one group Gi is nonmeasurable with respect to .
The proof of this theorem utilizes the classical result from the theory of groups, stating that every divisible commutative group can be represented as the direct sum of a family of groups each of which is isomorphic either to Q (the group of all rationals) or to the quasi-cyclic group of type where p is a prime number (see, e.g., [4]).
Obviously, in Theorem 2 any uncountable vector space over Q can be taken as (in particular, we may put where ). Also, we may put , where S denotes the one-dimensional torus and is an arbitrary nonzero cardinal.
Actually, each subgroup Gi of the preceding theorem turns out to be a -absolutely negligible subset of (see [3]). Therefore, for a given , every probability quasi-invariant measure on can be extended to a probability quasi-invariant measure satisfying the relation .
It would be interesting to extend Theorem 2 to a more general class of uncountable groups (not necessarily divisible or commutative). In this connection, let us remark that the assertion of this theorem fails to be true for some uncountable groups. In particular, if is uncountable and contains no proper uncountable subgroup, then the above-mentioned theorem is obviously false for . On the other hand, by starting with the result of this theorem, it is not difficult to construct an uncountable noncommutative nondivisible group with a countable family of its subgroups, such that, each is a -absolutely negligible set and, for any probability (left) quasi-invariant measure on , at least one Gi is nonmeasurable with respect to .
A statement similar to Theorem 2 can be established (under some additional set-theoretical hypotheses) for -finite diffused measures which are given on a commutative group of cardinality continuum and, in general, are not assumed to be quasi-invariant. More exactly, we have the following
Theorem 3. Suppose that the Continuum Hypothesis holds, and let be any commutative group of cardinality continuum. Then
there exists a countable family of subgroups of , such that, for each nonzero -finite diffused measure on , there exists at least one group Gi nonmeasurable with respect to .
The proof of this statement is based on well-known results concerning the algebraic structure of infinite commutative groups (see, e.g., [4]) and on some properties of the classical Banach-Kuratowski matrix whose existence is implied by the Continuum Hypothesis (see [1]). Let us remark that the assumption
of commutativity of a given group is essential in the formulation of Theorem 3. Indeed, under the Continuum Hypothesis, this theorem fails
to be true for some noncommutative groups of cardinality continuum. It would be interesting to extend Theorem 3 to a more general class of uncountable groups .
R E F E R E N C E S
1. S.Banach, K.Kuratowski, Sur une generalisation du probleme de la mesure, Fund. Math., vol. 14, 1929, pp. 127 - 131.
2. S.Kakutani, J.Oxtoby, Construction of a nonseparable invariant extension of the Lebesgue measure space, Ann. Math., vol. 52, 1950, pp. 580 - 590.
3. A.B.Kharazishvili, Invariant Extensions of Lebesgue Measure, Izd. Tbil. Gos. Univ., Tbilisi, 1983 (in Russian).
4. A.G.Kurosh, The Theory of Groups, Izd. Nauka, Moscow, 1967 (in Russian).
5. G.W.Mackey, Borel structures in groups and their duals, Trans. Amer. Math. Soc., vol. 85, 1957, pp. 134 - 169.
6. E.Szpilrajn (E.Marczewski), Sur l'extension de la mesure lebesguienne, Fund. Math., vol. 25, 1935, pp. 551 - 558.
Professor Alexander Kechris, Main Lecture, Session 6
Linear algebraic groups and descriptive set theory
Abstract: This is a joint work with Scot Adams.
In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invarients. Frequently, these objects can be viewed as elements of a Polish (complete separable metric) space X and the equivalence turns out to be a Borel equivalence relation E on X. A complete classification of X up to E consists of finding a set of invarients I and a map such that . For this to be of any interest, both I and c must be explicit or definable and as simple and concrete as possible. The theory of Borel equivalence relations studies the set theoretic nature of possible invarients and develops a mathematical framework for measuring the complexity of such classification problems.
In organizing this study, the following concept of reducibility is fundamental.
Let E,F be equivalece relations on Polish spaces X,Y respecively. We say
that E is Borel reducible to F, in symbols,
The structure of this partial order and the corresponding hierarchy of classification problems has been extensively studied over the last decade. In this talk, I will give an introduction to this theory and discuss the recent solution of a long standing problem in this area in which the ergodic theory of linear algebraic groups, and more particulrly the so-called superrigidity theory of R. Zimmer, plays a role.
Professor Emma D'Aniello , Lecture 43, Session 6
Cn functions, Hausdorff measures and analytic sets
Abstract: In this talk, we characterize in terms of Hausdorff measures and descriptive complexity those subsets M of the reals which are
Professor Janusz Pawlikowski, Lecture 44, Session 6
A combinatorial principle in the iterated perfect set model - further development
Abstract:
Professor Udayan B. Darji, Lecture 45, Session 6
Cn functions, Hausdorff measures and analytic sets
Abstract: In this talk, we characterize in terms of Hausdorff measures and descriptive complexity those subsets M of the reals which are
Professor T. H. Steele, Lecture 46, Session 5
Notions of stability for one-dimensional dynamical systems
Abstract: At the Twentieth Summer Symposium in Real Analysis, A. M. Bruckner posed several questions regarding the iterative stability of continuous functions as they undergo small perturbations, as well as why these questions are of general interest B. In particular, how are the set of -limit points and the collection of -limit sets of a function affected by slight changes in that function? As Bruckner discusses in B, we may also want to ask these questions when restricting our attention to particular subsets of C(I,I), such as those functions that are in some way nonchaotic, or those functions that satisfy a particular smoothness condition. As one sees from various examples found in B and TH, in general, both the set of -limit points and the collection of -limit sets of a typical function are affected dramatically by arbitrarily small ! perturbations. In TH2QTSNlabelTH2 we make some progress towards understanding the continuity structure of the maps and . In particular, we are able to characterize those functions at which given by is continuous, as well as characterize the points of continuity of the map given by when we resrict the domain of to those continuous functions possessing zero topological entropy.
Professor Marek Balcerzak, Lecture 47, Session 6
On Marczewski-Burstin representations of certain algebras of sets
Abstract: We study a general problem how to represent a given algebra of sets in a wayanalogous to that used by Marczewski and Burstin. We show that severalwell-known algebras on the real line are MB-representable. One of the resultsstates that, under GCH, the algebra of Borel sets is MB-representable by a familyof non-Borel sets. Also under GCH, an example of algebras that are notMB-representable is constructed. We discuss several examples concerning outer and inner MB-representations of algebras.
Professor Eric Talvila, Lecture 48, Session 6
Some divergent integrals and the Riemann-Lebesgue lemma
Abstract:
While looking through integral tables I found some Fourier integrals that have been tabulated for many years but diverge. We will trace them back to an original error made by a(shockingly famous) mathematician. (Not someone at this conference) They lead one to think of the Riemann-Lebesgue lemma. For an L1 function this says that its Fourier transform tends to zero at infinity. If we consider conditionally convergent integrals we get dramatically different results.