414/514 Homework 2 – Monotone and Baire one functions

This set is due in three weeks, on Monday, November 3, at the beginning of lecture.

1. Let be increasing. We know that and exist for all , and that has at most countably many points of discontinuity, say For each let be the intervals and . Some of these intervals may be empty, but for each at least one of them is not. (Here we follow the convention that and .) Let denote the length of the interval , and say that an interval precedes a point iff .

Verify that and, more generally, for any ,

precedes precedes .

Define a function by setting . Show that is increasing and continuous.

Now, for each , define so that , , and for all . Show that each is increasing, and its only discontinuity points are .

Prove that uniformly.

Use this to provide a (new) proof that increasing functions are in Baire class one.

2. Solve exercise 3.Q in the van Rooij-Schikhof book: If is such that for all , we have that and exist, then is the uniform limit of a sequence of step functions. The approach suggested in the book is the following:

Show that it suffices to argue that for every there is a step function such that for all .

To do this, consider the set there is a step function on such that for all .

Show that is non-empty. Show that if and , then also . This shows that is an interval or , with . Show that in fact the second possibility occurs, that is, . For this, the assumption that exists is useful. Finally, show that . For this, use now the assumption that exists.

3.(This problem is optional.) Find a counterexample to the following statement: If is the pointwise limit of a sequence of functions , then there is a dense subset where the convergence is in fact uniform. What if and the functions are continuous? Can you find a (reasonable) weakening of the statement that is true?

4. (This is example 1.1 in Andrew Bruckner’s Differentiation of real functions, CRM monograph series, AMS, 1994. MR1274044(94m:26001).) We want to define a function . Let be the Cantor set in . Whenever is one of the components of the complement of , we define for . For not covered by this case, we define . Verify that is a Darboux continuous function, and that it is discontinuous at every point of .

Verify that is not of Baire class one, but that there is a Baire class one function that coincides with except at (some of) the endpoints of intervals as above.

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3 Responses to 414/514 Homework 2 – Monotone and Baire one functions

I have corrected the definition of the function in problem 1. Thanks to Jeremy Siegert for noticing the typo in the original version, and for noting that an should be .

Thanks to Stuart Nygard for noticing a further typo in question 2 (some should have been s). Fixed now.

In problem 1 we are supposed to show that each is discontinuous on the points . There is no based on how we indexed ‘s points of discontinuity, but it looks as though is discontinuous at . Should it be that each is discontinuous on ?

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${ […]

Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]

Equality is part of the background (first-order) logic, so it is included, but there is no need to mention it. The situation is the same in many other theories. If you want to work in a language without equality, on the other hand, then this is mentioned explicitly. It is true that from extensionality (and logical axioms), one can prove that two sets are equ […]

$L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZF}$) well-orders all of $L$, so that its restriction to any specific set $A$ in $L$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction […]

Gödel sentences are by construction $\Pi^0_1$ statements, that is, they have the form "for all $n$ ...", where ... is a recursive statement (think "a statement that a computer can decide"). For instance, the typical Gödel sentence for a system $T$ coming from the second incompleteness theorem says that "for all $n$ that code a proof […]

When I first saw the question, I remembered there was a proof on MO using Ramsey theory, but couldn't remember how the argument went, so I came up with the following, that I first posted as a comment: A cute proof using Schur's theorem: Fix $a$ in your semigroup $S$, and color $n$ and $m$ with the same color whenever $a^n=a^m$. By Schur's theo […]

It depends on what you are doing. I assume by lower level you really mean high level, or general, or 2-digit class. In that case, 54 is general topology, 26 is real functions, 03 is mathematical logic and foundations. "Point-set topology" most likely refers to the stuff in 54, or to the theory of Baire functions, as in 26A21, or to descriptive set […]

I have corrected the definition of the function in problem 1. Thanks to Jeremy Siegert for noticing the typo in the original version, and for noting that an should be .

Thanks to Stuart Nygard for noticing a further typo in question 2 (some should have been s). Fixed now.

In problem 1 we are supposed to show that each is discontinuous on the points . There is no based on how we indexed ‘s points of discontinuity, but it looks as though is discontinuous at . Should it be that each is discontinuous on ?

Yes, exactly.