Egorov’s Theorem

General overview/ idea

Given a sequence of measurable functions that converge pointwise, we can’t say that they converge uniformly. However, if the total measure of the space is finite, then we can always restrict the domain (while taking out an arbitrarily small set) to where the sequence does converge uniformly.

Theorem

Given a measure space $(X,\mathcal{S},\mu )$ where $\mu (X)<\infty$ and a sequence of $\mathcal{S}$ -measureable functions $f_1,f_2,\cdots :X\to \mathbb{R}$ converges pointwise to a function $f$.

$\forall ϵ>0,\exists$ a set $E\in \mathcal{S}$ such that $\mu (X\backslash E)<ϵ$ and $f_1,f_2,\dots$ converges uniformly on $X\backslash E$

Proof

General idea:

Uniform convergence means that our choice of $ϵ$ doesn’t depend on the value of x. Instead, let the set E depend on x, then the uniform convergence can be guaranteed on the set $X\backslash E$

I don’t want to write the entire thing out, it is pretty long. In its full glory, the proof is on @axler2019 page 64.