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.

Verify that is in Baire class two.

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Friday, October 10th, 2014 at 12:45 pm and is filed under 414/514: Analysis I. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

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 ?

The technique of almost disjoint forcing was introduced in MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. Some applications of almost disjoint sets. In Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), pp. 84–104, North-Holland, Amsterdam, 1970. Fix an almost disjoint family $X=(x_\alpha:\alpha

At the moment most of those decisions come from me, at least for computer science papers (those with a 68 class as primary). The practice of having proceedings and final versions of papers is not exclusive to computer science, but this is where it is most common. I've found more often than not that the journal version is significantly different from the […]

The answer is no in general. For instance, by what is essentially an argument of Sierpiński, if $(X,\Sigma,\nu)$ is a $\sigma$-finite continuous measure space, then no non-null subset of $X$ admits a $\nu\times\nu$-measurable well-ordering. The proof is almost verbatim the one here. It is consistent (assuming large cardinals) that there is an extension of Le […]

I assume by $\aleph$ you mean $\mathfrak c$, the cardinality of the continuum. You can build $D$ by transfinite recursion: Well-order the continuum in type $\mathfrak c$. At stage $\alpha$ you add a point of $A_\alpha$ to your set, and one to its complement. You can always do this because at each stage fewer than $\mathfrak c$ many points have been selected. […]

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 negative. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${\mathfrak c}$ (This doesn't matter, all we need is that it is strictly larger. T […]

Yes, the suggested rearrangement converges to 0. This is a particular case of a result of Martin Ohm: For $p$ and $q$ positive integers rearrange the sequence $$\left(\frac{(−1)^{n-1}} n\right)_{n\ge 1} $$ by taking the ﬁrst $p$ positive terms, then the ﬁrst $q$ negative terms, then the next $p$ positive terms, then the next $q$ negative terms, and so on. Th […]

Yes, by the incompleteness theorem. An easy argument is to enumerate the sentences in the language of arithmetic. Assign to each node $\sigma $ of the tree $2^{

A simple example is the permutation $\pi$ given by $\pi(n)=n+2$ if $n$ is even, $\pi(1)=0$, and otherwise $\pi(n)=n−2$. It should be clear that $\pi$ is computable and has the desired property. By the way, regarding the footnote: if a bijection is computable, so is its inverse, so $\pi^{-1}$ is computable as well. In general, given a computable bijection $\s […]

The question is asking to find all polynomials $f$ for which you can find $a,b\in\mathbb R$ with $a\ne b$ such that the displayed identity holds. The concrete numbers $a,b$ may very well depend on $f$. A priori, it may be that for some $f$ there is only one pair for which the identity holds, it may be that for some $f$ there are many such pairs, and it may a […]

The reflection principle is a theorem schema in ZFC, meaning that for each formula $\phi(\vec x)$ we can prove in ZFC a version of the principle for $\phi$. In particular, it gives us that if $\phi$ holds (in the universe of sets) then there is some ordinal $\alpha$ such that $V_\alpha\models \phi$. It follows from this that (assuming its consistency) $\math […]

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.