next up previous
Next: New classic-like results Up: Set Theoretic Real Analysis Previous: Historical background

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 tex2html_wrap_inline5101 is measurable then the iterated integrals tex2html_wrap_inline5103 and tex2html_wrap_inline5105 exist and are both equal to the double integral tex2html_wrap_inline5107, where tex2html_wrap_inline5109 stands for the Lebesgue measure on tex2html_wrap_inline5111. 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 tex2html_wrap_inline5121 be a well ordering of [0,1] in order type continuum tex2html_wrap_inline5125 and define tex2html_wrap_inline5127. Let f be the characteristic function tex2html_wrap_inline5131 of A. Then for every fixed tex2html_wrap_inline5135 the set tex2html_wrap_inline5137 is an initial segment of a set ordered in type tex2html_wrap_inline5125. So, by CH, it is at most countable and
Similarly, for each tex2html_wrap_inline5141 the set tex2html_wrap_inline5143 is at most countable and
Thus, tex2html_wrap_inline5145. tex2html_wrap_inline5147

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 tex2html_wrap_inline5075 of cardinality less than continuum has measure zero, i.e., when the smallest cardinality tex2html_wrap_inline5159 of the non-measurable subset of tex2html_wrap_inline5075 is equal to tex2html_wrap_inline5125. Since the equation tex2html_wrap_inline5165 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 tex2html_wrap_inline5131; (B) there is no set tex2html_wrap_inline5169 with tex2html_wrap_inline5171 satisfying Theorem 2.2 if tex2html_wrap_inline5173, where tex2html_wrap_inline5175 is the smallest cardinality of a covering of tex2html_wrap_inline5075 by the sets of measure zero. (It is well known that the inequality tex2html_wrap_inline5173 is consistent with ZFC.)

A discussion of a similar problem for the functions tex2html_wrap_inline5181 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 tex2html_wrap_inline5185 and tex2html_wrap_inline5187 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 tex2html_wrap_inline5197 with any measure zero set N is at most countable.gif Another set that satisfies the conclusion of Theorem 2.3, known as Luzin set (see [126] or [127, property Ctex2html_wrap_inline5201,]), is defined as an uncountable subset L of tex2html_wrap_inline5075 whose intersection tex2html_wrap_inline5207 with any meager set M is at most countable.gif 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 Gtex2html_wrap_inline5215 measure zero sets (Ftex2html_wrap_inline5217 meager sets) as tex2html_wrap_inline5219 and define S (L, respectively) as a set tex2html_wrap_inline5225 where tex2html_wrap_inline5227. The choice is possible since, by CH, the family tex2html_wrap_inline5229 is at most countable implying that its union is not equal to tex2html_wrap_inline5075.

It is also easy to see that this construction can be carried out if tex2html_wrap_inline5233 (and its category analog tex2html_wrap_inline5235 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 tex2html_wrap_inline5125. Since many models of ZFC satisfy either tex2html_wrap_inline5233 or tex2html_wrap_inline5235 (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 tex2html_wrap_inline5075 can be extended to a continuous function on a Gtex2html_wrap_inline5215 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 tex2html_wrap_inline5315 less than tex2html_wrap_inline5125.

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 tex2html_wrap_inline5323. 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 tex2html_wrap_inline5355 be a partition of tex2html_wrap_inline5075 such that tex2html_wrap_inline5359 is a dense Gtex2html_wrap_inline5215 set of measure zero and tex2html_wrap_inline5363 is nowhere dense for each n>0. Define tex2html_wrap_inline5367 by putting f(x)=n for tex2html_wrap_inline5371. Now, tex2html_wrap_inline5373 is discontinuous for any dense tex2html_wrap_inline5375 which is nowhere measure zero.

Indeed, if tex2html_wrap_inline5375 is dense and nowhere measure zero then there exists tex2html_wrap_inline5379. Now, if every open set U containing x intersects tex2html_wrap_inline5385 for infinitely many n then tex2html_wrap_inline5373 is discontinuous at x. Otherwise, there is an open set U containing x and intersecting only finitely many tex2html_wrap_inline5363's. So, we can find a non-empty open interval tex2html_wrap_inline5399 such that tex2html_wrap_inline5401. But this means that tex2html_wrap_inline5403 has measure zero, a contradiction. tex2html_wrap_inline5147

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 tex2html_wrap_inline5367 is symmetrically continuous at tex2html_wrap_inline5413 if
and f is approximately symmetrically differentiable at x if there exists a set tex2html_wrap_inline5419 such that x is a (Lebesgue) density point of tex2html_wrap_inline5423 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 tex2html_wrap_inline5431. We will say that f has a co-countable symmetric derivative at x and denote it by tex2html_wrap_inline5437 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 formgif from which it follows immediately that the theorem remains true under MA, if the co-countable symmetric derivatives tex2html_wrap_inline5437 are replaced by the approximate symmetric derivatives tex2html_wrap_inline5463. However, neither Theorem 2.13 nor its version with tex2html_wrap_inline5463 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 tex2html_wrap_inline5173. (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.gif 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 tex2html_wrap_inline5367 is weakly continuous  at x if there are sequences tex2html_wrap_inline5501 and tex2html_wrap_inline5503 such that
This notion is so weak that it is impossible to find a function tex2html_wrap_inline5367 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 tex2html_wrap_inline5367 is weakly symmetrically continuous at x if there is a sequence tex2html_wrap_inline5513 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 tex2html_wrap_inline5529 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 tex2html_wrap_inline5557 and there exists tex2html_wrap_inline5559 such that each tex2html_wrap_inline5529 has at most tex2html_wrap_inline5563 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 tex2html_wrap_inline5073:

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 tex2html_wrap_inline5073 such that no three vertices a,b,x spanning isosceles triangle belong to the same element of the partition.

next up previous
Next: New classic-like results Up: Set Theoretic Real Analysis Previous: Historical background