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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