Mostly to take a break from marking exams, I thought I'd start a new recurring series here about mathematics papers and books that I find, both new and old. The "new" will consist mainly of preprints that look interesting (to encourage me to browse the arXiv) and the "old" will consists of papers I will likely read (to encourage me to read, or at least skim, more papers).

New Preprints

  1. Clark Barwick, "On the algebraic K-theory of higher categories": For a while I've wanted to learn a bit about higher category theory, but I've not yet found any application or motivation for it in something I'm already really interested in (including the stuff here). Perhaps this paper by Barwick will change my mind: algebraic K-theory may be best viewed as a functor with a universal property that generalises the well-known universal properties known for the classical K groups.
  2. Booker, Hiary, and Keating, Detecting squarefree numbers": Under the Generalised Riemann Hypothesis the authors propose an algorithm to test whether an integer is squarefree, without needing the number's factorisation.
  3. Shalit, "A sneaky proof of the maximum modulus principle": This is a proof of the maximum modulus principle in complex analysis, now with even less complex analysis! This paper also appeared in the American Mathematical Monthly and the author wrote a blog post about it as well.

View the whole post to reveal the hidden classic:
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Posted by Jason Polak on 27. August 2011 · Write a comment · Categories: measure-theory

In the previous post, we saw how to use a basic theorem on Lebesgue points to prove that if $ A$ and $ B$ are measurable subsets of the real line with positive measure, then $ A+B$ contains an interval. We shall continue now to prove this again using a different, less involved method. This solution is based on a hint in Problem 19 of Chapter 9 in Rudin's book.

The Proof

Recall that we are proving: if $ A$ and $ B$ are measurable sets of real numbers such that $ A$ and $ B$ have positive measure, then $ A+B$ contains an interval.
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Posted by Jason Polak on 27. August 2011 · Write a comment · Categories: measure-theory · Tags: , ,

Today I shall continue in the spirit of my last post, which was essentially a revised set of notes on material for my qualifying exam. Here, and in the next post, we shall see two ways to prove that if $ A$ and $ B$ are Lebesgue-measurable subsets of the real line with positive measure, then $ A+B$ contains an interval. The notation $ A+B$ means $ \{ a + b : a\in A, b\in B\}$.

Before we start, let us examine counterexamples. Firstly, the converse to the statement is not true. I'll leave it as an exercise to show that the Cantor set $ C$ is such that $ C+C$ contains an interval, yet $ C$ has measure zero. On the other hand, $ \mathbb{Q}+\mathbb{Q}=\mathbb{Q}$, and $ \mathbb{Q}$ does not contain an interval. In fact, if we prove the above then we shall have another proof that $ \mathbb{Q}$ has measure zero, although this follows more directly by observing that $ \mathbb{Q}$ is countable, and that points have measure zero.

Both these solutions were hinted at in Rudin's "Real and Complex Analysis". I've broken up these solutions into two posts for convenience. In the sequel, measurable means Lebesgue measurable on the real line, and $ m$ denotes the Lebesgue measure.
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Posted by Jason Polak on 20. August 2011 · 2 comments · Categories: measure-theory

For my qualifying exam next week, I made a few notes on the fundamental theorem of calculus in the Lebesgue setting and I've decided to post them in case they might be of use to someone else. I shall sketch the proof and try to explain the main points, aiming for a broad overview. The interested reader should consult Chapter 7 of Rudin's Real and Complex Analysis for full proofs.

In the Riemann setting, for a continuous Riemann-integrable function $ f:[a,b]\to\mathbb{R}$ differentiable on $ (a,b)$, one form of the fundamental theorem is that

$ \int_a^x f'(t)dt = f(x) – f(a)$.

Here the integral is the Riemann integral. Now, one can make various technical modifications to the hypotheses of the usual analysis ilk, such as allowing $ f$ to be differentiable on $ (a,b)$ except for some finite set, but the above statement seems to capture the essence of the fundamental theorem. Now, the proof of the fundamental theorem is short and sweet: use the definition of the Riemann integral as a limit of Riemann sums and apply the mean value theorem.

We would like a similar statement to hold for the Lebesgue measure $ \mu$ on the real line.
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