## Lusin’s Theorem and Egorov’s Theorem

Lusin’s Theorem and Egorov’s Theorem are the second and third of Littlewood’s famous Three Principles.

There are many variations and generalisations, the most basic of which I think are found in Royden’s book.

## Lusin’s Theorem:

Informally, “every measurable function is nearly continuous.”

(Royden) Let $f$ be a real-valued measurable function on $E$. Then for each $\epsilon>0$, there is a continuous function $g$ on $\mathbb{R}$ and a closed set $F\subseteq E$ for which $\displaystyle f=g\ \text{on}\ F\ \text{and}\ m(E\setminus F)<\epsilon.$

## Egorov’s Theorem

Informally, “every convergent sequence of functions is nearly uniformly convergent.”

(Royden) Assume $m(E)<\infty$. Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise on $E$ to the real-valued function $f$.

Then for each $\epsilon>0$, there is a closed set $F\subseteq E$ for which $\displaystyle f_n\to f\ \text{uniformly on}\ F\ \text{and}\ m(E\setminus F)<\epsilon.$