Category Archives: measure-theory

A set that is not Lebesgue measurable

The Lebesgue measure $\lambda$ on the real line is a countably additive measure that assigns to each interval $[a,b]$ with $a \leq b$ its length $b-a$. Why construct the Lebesgue measure? It’s so that we can get around a blemish of the Riemann integral: namely, that the Riemann integration theory does not know how to […]

Preprints and Classics 1: Higher cats, squarefree, max modulus

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 […]

The Sumset of Sets of Positive Measure, Continued

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 […]

The Sumset of Sets of Positive Measure

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, […]

Fundamental Theorem of Calculus, Lebesgue Version

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 […]