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.

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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 ?

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

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 […]

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]

Yes, there is an $\aleph_2$ and an $\aleph_3$, and there are alephs beyond all the $\aleph_n$. A suitable version of Cantor's diagonal proof is perfectly general and shows that, for any set $X$, $|X|

Given a class $S$, to say that it can be proper means that it is consistent (with the axioms under consideration) that $S$ is a proper class, that is, there is a model $M$ of these axioms such that the interpretation $S^M$ of $S$ in $M$ is a proper class in the sense of $M$. It does not mean that $S$ is always a proper class. In fact, it could also be consis […]

As the other answers point out, the question is imprecise because of its use of the undefined notion of "the standard model" of set theory. Indeed, if I were to encounter this phrase, I would think of two possible interpretations: The author actually meant "the minimal standard model of set theory", that is, $L_\Omega$ where $\Omega$ is e […]

I assume you want the measures to be $0$ on singletons. If this is the intention, it is impossible for both measures to coexist, for the reason that you identify: $m_1$ would force $\kappa$ to be (atomlessly) real-valued measurable, which implies that $\kappa\le\mathfrak c=|\mathbb R|$, while $m_2$ would force $\kappa$ to be a measurable cardinal, and theref […]

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.