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# New developments in classical problems

The first problem we wish to mention here is connected with the Fubini-Tonelli Theorem. The theorem says, in particular, that if a function is measurable then the iterated integrals and exist and are both equal to the double integral , where stands for the Lebesgue measure on . But what happens when f is non-measurable? Clearly, then the double integral does not exist. However, the iterated integrals might still exist. Must they be equal? The next theorem, which is a classical example of an application of the Continuum Hypothesis in real analysis, gives a negative answer to this question.

PROOF. Let be a well ordering of [0,1] in order type continuum and define . Let f be the characteristic function of A. Then for every fixed the set is an initial segment of a set ordered in type . So, by CH, it is at most countable and

Similarly, for each the set is at most countable and

Thus, .

Sierpinski's use of the Continuum Hypothesis in the construction of such a function begs the question whether such a function can be constructed using only the axioms of ZFC. The negative answer was given in the 1980's by Laczkovich [87], Friedman [60] and Freiling [54], who independently proved the following theorem.

It is also worthwhile to mention that the function f from the proof of Theorem 2.2 has the desired property as long as every subset of of cardinality less than continuum has measure zero, i.e., when the smallest cardinality of the non-measurable subset of is equal to . Since the equation holds in many models of ZFC in which CH fails (for example, it is implied by MA) Theorem 2.2 is certainly not equivalent to CH. On the other hand, Laczkovich proved Theorem 2.2 by noticing that: (A) the existence of an example as in the statement of Theorem 2.2 implies the existence of such an example as in its proof, i.e., in form of ; (B) there is no set with satisfying Theorem 2.2 if , where is the smallest cardinality of a covering of by the sets of measure zero. (It is well known that the inequality is consistent with ZFC.)

A discussion of a similar problem for the functions and the n-times iterated integrals can be found in a 1990 paper of Shipman [123]. The same paper contains also two easy ZFC examples of measurable functions and for which the iterated integrals exist but are not equal. Thus, the restriction of the above problem to the non-negative functions is essential.

Another classical result arises from a different theorem of Sierpinski of 1928.

The set S from the original proof of Theorem 2.3 is called Sierpinski set and it has the property that its intersection with any measure zero set N is at most countable. Another set that satisfies the conclusion of Theorem 2.3, known as Luzin set (see [126] or [127, property C,]), is defined as an uncountable subset L of whose intersection with any meager set M is at most countable. The existence of a Luzin set is also implied by CH. In fact, the constructions of sets S and L under the assumption of CH are almost identical: you list all G measure zero sets (F meager sets) as and define S (L, respectively) as a set where . The choice is possible since, by CH, the family is at most countable implying that its union is not equal to .

It is also easy to see that this construction can be carried out if (and its category analog in case of construction of L). The sets constructed that way are called generalized Sierpinski and Luzin sets, respectively, and they also satisfy the conclusion of Theorem 2.3 independently of the size of . Since many models of ZFC satisfy either or (for example, both conditions are implied by MA) is has been a difficult task to find a model of ZFC in which the conclusion of Theorem 2.3 fails. It has been found by A. W. Miller in 1983.

In his proof of Theorem 2.4 Miller used the iterated perfect set model, which will be mentioned in this paper in several other occasions.

Some of the most recent set-theoretic results concerning classical problems in real functions are connected with a theorem of Blumberg from 1922.

The set D constructed by Blumberg is countable. In a quest whether it can be chosen any bigger Sierpinski and Zygmund proved in 1923 the following theorem.

Theorem 2.6 immediately implies the following corollary, which shows that there is no hope for proving in ZFC a version of the Blumberg theorem in which the set D is uncountable.

The proof of Theorem 2.6 is a straightforward transfinite induction diagonal argument after noticing that every continuous partial function on can be extended to a continuous function on a G set.

Corollary 2.7 raises the natural question about the importance of the assumption of CH in its statement. Is it consistent that the set D in Blumberg Theorem can be uncountable? Can it be of positive outer measure, or non-meager?

The cardinality part of these questions is addressed by the following theorem of Baldwin from 1990.

Thus under MA the size of the set D is clear. By Theorem 2.6 it cannot be chosen of cardinality continuum (at least for some functions), but it can be always chosen of any cardinality less than .

One might still hope to be able to prove in ZFC that for any f the set D can be found of an arbitrary cardinality . However, this is false as well, as noticed by Shelah in his paper from 1995.

The category version of a question on a size of D has been also settled by Shelah in the same paper.

The measure version of the question is less clear. It has been noticed by J. Brown in 1977 that the precise measure analog of Theorem 2.10 cannot be proved. (This has been also noticed independently by K. Ciesielski, whose proof is included below.)

PROOF. Let be a partition of such that is a dense G set of measure zero and is nowhere dense for each n>0. Define by putting f(x)=n for . Now, is discontinuous for any dense which is nowhere measure zero.

Indeed, if is dense and nowhere measure zero then there exists . Now, if every open set U containing x intersects for infinitely many n then is discontinuous at x. Otherwise, there is an open set U containing x and intersecting only finitely many 's. So, we can find a non-empty open interval such that . But this means that has measure zero, a contradiction.

However, the following problem asked by Heinrich von Weizsäcker [59, Problem AR(a),] remains open.

Other generalizations of Blumberg's theorem can be also found in a 1994 survey article [11]. (See also recent papers [12] and [68].)

In the past few years a lot of activity in real analysis was concentrated around symmetric properties of real functions. (See Thomson [138].) Recall that a function is symmetrically continuous at if

and f is approximately symmetrically differentiable at x if there exists a set such that x is a (Lebesgue) density point of and that the following limit exists

This limit, which does not depend on the choice of a set S, is called the approximate symmetric derivative of f at x and is denoted by . We will say that f has a co-countable symmetric derivative at x and denote it by if the set S in the above definition can be chosen to be countable.

One of the long standing conjectures (with several incorrect proofs given earlier, some even published) was settled by Freiling and Rinne in 1988 by proving the following theorem.

The importance of the measurability assumption in Theorem 2.12 was long known from the following theorem of Sierpinski of 1936.

In fact, in [128] Theorem 2.13 is stated in a bit stronger form from which it follows immediately that the theorem remains true under MA, if the co-countable symmetric derivatives are replaced by the approximate symmetric derivatives . However, neither Theorem 2.13 nor its version with can be proved in ZFC. This follows from the following two theorems of Freiling from 1990.

Thus the existence of a function as in Theorem 2.13 is in fact equivalent to the Continuum Hypothesis.

More precisely, Freiling proves that the conclusion of Theorem 2.15 follows the property that is just a bit stronger than the inequality . (Compare comment following Theorem 2.2.)

Another direction in which the symmetric continuity research went was the study of how far symmetric continuity can be destroyed. First note that clearly every continuous function is symmetrically continuous, but not vice versa, since the characteristic function of a singleton is symmetrically continuous. However, it is not difficult to find functions which are nowhere symmetrically continuous. For example, the characteristic function of any dense Hamel basis is such a function. How much more can we destroy symmetric continuity?

In the non-symmetric case probably the weakest (bilateral) version of continuity that can be defined is the following. A function is weakly continuous  at x if there are sequences and such that

This notion is so weak that it is impossible to find a function which is nowhere weakly continuous. This follows from the following easy, but a little surprising theorem.

A natural symmetric counterpart of weak continuity is defined as follows. A function is weakly symmetrically continuous at x if there is a sequence such that

However, the symmetric version of Theorem 2.16 badly fails: there exist nowhere weakly symmetrically continuous functions (which are also called uniformly antisymmetric functions). Their existence follows immediately from the following theorem of Ciesielski and Larson from 1993.

The function f from Theorem 2.17 raises the questions in two directions. Can the range of f be any smaller? Can the size of all sets be uniformly bounded? The first of this questions leads to the following open problem from [33]. (See also problems listed in [138].)

Concerning part (a) of this problem it has been proved in 1993 by Ciesielski [25] that the range of uniformly antisymmetric function must have at least 4 elements. (Compare also [27].)

The estimation of sizes of sets from Theorem 2.17 has been examined by Komjáth and Shelah in 1993, leading to the following two theorems.

Theorem 2.18 suggests that the converse of Theorem 2.19 should also be true. However, this is still unknown, leading to another open problem.

For k=0 the positive answer is implied by Theorem 2.18. Also, it is consistent that and there exists such that each has at most elements. This follows from another theorem of Komjáth and Shelah [84, Thm 1,]. (See also a paper [28] of Ciesielski related to this subject.)

In fact, the proof of Theorem 2.17 gives also the following version for functions on :

• There exists a function such that the set

is finite for every .
This statement is related to the following recent theorem of J. Schmerl, which solves a long standing problem of Erdos [101, Problem 15.9,]. (See also a survey article [82] for more on this problem.)

Thus, this theorem says, that there exists (in ZFC) a countable partition of such that no three vertices a,b,x spanning isosceles triangle belong to the same element of the partition.

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