**REAL ANALYSIS, XXIV:
**

Professor Marianna Csornyei, Lecture 1, Session 1

*Absolutely continuous functions of several variables*

For a fixed
symmetric convex set of non-empty interior
let
denote the set of all `balls' of *R*^{n} in the norm defined by
*K*_{0},
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 *R*^{n}, 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*

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 **Z**_{2} 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*

Professor Robert Vallin, Lecture 3, Session 1
*Cantor Sets Thickness, and Porosity*

Professor Chris Ciesielski, Lecture 4, Session 1

*On anti-Schwartz functions and functions with two element range*

A function
is
antisymmetric (anti-Schwartz) at
provided there is no sequence such that
*f*(*x*+*h*_{n})=*f*(*x*-*h*_{n}) (
*f*(*x*+*h*_{n})=*f*(*x*-*h*_{n})=*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 *h*_{n}, 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*

Professor Peter Wingren, Lecture 6, Session 1

*Smooth Functions Defined on Fractal Sets*

Professor Krzysztof Plotka, Lecture 7, Session 1

*Sum of Sierpinski-Zygmunt and almost continuous functions*

**SESSION 2, WEDNESDAY AFTERNOON**

Professor Slava V. Chistyakov, Lecture 08, Session 2

*On mappings of finite generalized variation and nonlinear operators*

Professor Kenneth Falconer, Main Lecture, Session 2

*Nonlinear Mercerian Theorems*

Professor Maarit Jarvenpaa, Lecture 09, Session 2

*Porosities of measures*

Professor Henry Fast, Lecture 10, Session 2

*All translates of perfect in all directions sets meet Cantor-like products.*

Professor Dave Lawrence Renfro, Lecture 11, Session 2

*Mean Porous Sets*

Marton Elekes, Lecture 12, Session 2

*Linearly ordered sets of real functions*

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*

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*

Professor Kiko Kawamura, Lecture 15, Session 2

*Computational Complexity of Self-similar sets*

Professor Maria Eugenia Mera Rivas, Lecture 16, Session 2

*A Zero-One Half Law for Porosity of Measures*

Dr. Roy Mimna, Lecture 17, Session 2

*Persistence stability of non-invertible maps on compact metric spaces*

Professor Francis Jordan, Lecture 18, Session 2

*Blumberg theorem for sets of functions*

**SESSION 3, THURSDAY MORNING**

Professor Siegfried Graf, Lecture 19, Session 3

*Asymptotics in quantization of random vectors*

Professor Yuval Peres, Main Lecture, Session 3

*When Projections Reduce Dimension What 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:

- 1.
- For any Borel set
*E*in*R*^{3}with*dim E*>2, for a.e. line*L*through the origin, the orthogonal projection of*E*to*L*has nonempty interior. - 2.
- For any Borel set
*A*in*R*^{2}with Hausdorff dimension*dim A*>3/2, there are points*x*in*A*such that the ``pinned distance set'' has positive one-dimensional Lebesgue measure. - 3.
- The main application is to Bernoulli convolutions, a class of measures with connections to harmonic analysis, algebraic numbers, and dynamical systems.

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*

Professor Kandasamy Muthuvel, Lecture 21, Session 3

*Darboux fuction and continuous function*

Professor Ondrej Zindulka, Lecture 22, Session 3

*A set that is fat and slim at the same time*

Professor Svetlana V. Butler, Lecture 23, Session 3

*Approximation in the space of quasi-measures*

Professor Michal Rams, Lecture 24, Session 3

*Packing dimension estimate for the exceptinal parameter set*

Professor Matteo Rocca, Lecture 25, Session 3

*Smoothness conditions and differentiability properties of real functions*

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 *

Professor Martina Zaehle, Lecture 27, Session 4

*Riesz potentials of fractal measures*

Professor Jean-Pierre Kahane, Main Lecture, Session 4

*Baire category theorem and trigonometric series*

Professor Oleg Kovrijkine, Lecture 28, Session 4

*On periodizations of functions in higher dimensions*

Professor Giuseppa Riccobono , Lecture 29, Session 4

*A note on the PU-integral on an abstract space*

Professor Michal Morayne, Lecture 30, Session 4

*Martingales and Strong Differentiation of Integrals*

Professor Ralph E. Svetic, Lecture 31, Session 4

*Fixed points and the Composition of Darboux Baire One Functions*

Professor Stanley C. Williams, Lecture 32, Session 4

*Positivity of disintegration kernals of random measures generated by cascading exchangeable processes.*

Professor Grigore Ciurea, Lecture 33, Session 4

*On the Henstok and McShane integrability*

**SESSION 5, FRIDAY MORNING**

Professor Jaroslav Smital, Lecture 34, Session 5

*Minimal sets of continuous maps of the interval and distributional chaos.*

Professor Michal Misiurewicz, Main Lecture, Session 5

*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*

Professor Luisa Di Piazza , Lecture 36, Session 5

*A characterization of variationally McShane integrable Banach-space valued functions*

Professor Tomasz Natkaniec, Lecture 37, Session 5

*Universally Kuratowski-Ulam spaces*

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*

Professor Eduard Belinsky, Lecture 39, Session 5

*Metric entropy of subsets of absolutely convergent Fourier series*

Professor Jolanta Wesolowska, Lecture 40, Session 5

*Investigation of sets of convergence points of sequences of some real functions*

Dr. Bill Beyer, Lecture 41, Session 5

*History of the ham sandwich theorem: Steinhaus to the internet*

**SESSION 6, SATURDAY MORNING**

Professor Alexander Kharazishvili , Lecture 42, Session 6

*On measurability properties of subgroups of a given group*

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
*
*G*_{1},*G*_{2},...,*G*_{n}* 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
*H*_{1},*H*_{2},...,*H*_{k} are any subgroups of satisfying the
relations

then, for the group , the relation is valid, too, and each (left) quasi-invariant extension of , such that

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 **G*_{i}* 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 *G*_{i} 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 *G*_{i} 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 **G*_{i}* 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*

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,

if there is a Borel map such that

This ismply means that any complete invarients for

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
*C*^{n}* functions, Hausdorff measures and analytic sets*

- 1.
- the image under some
*C*^{n}function*f*of the set of points where deritvatives of first*n*orders are zero, - 2.
- the set of points where the level sets of some
*C*^{n}function is perfect, and - 3.
- the set of points where the level set of some
*C*^{n}function is uncountable

Professor Janusz Pawlikowski, Lecture 44, Session 6

*A combinatorial principle in the iterated perfect set model - further development*

Professor Udayan B. Darji, Lecture 45, Session 6
*C*^{n}* functions, Hausdorff measures and analytic sets*

- 1.
- the image under some
*C*^{n}function*f*of the set of points where deritvatives of first*n*order are zero, - 2.
- the set of points where the level sets of some
*C*^{n}function is perfect, and - 3.
- the set of points where the level set of some
*C*^{n}function is uncountable

Professor T. H. Steele, Lecture 46, Session 5

*Notions of stability for one-dimensional dynamical systems*

Professor Marek Balcerzak, Lecture 47, Session 6

*On Marczewski-Burstin representations of certain algebras of sets*

Professor Eric Talvila, Lecture 48, Session 6

*Some divergent integrals and the Riemann-Lebesgue lemma *

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 *L*^{1} function this says
that its Fourier transform tends to zero at infinity.
If we consider conditionally convergent integrals we get
dramatically different results.