Lusin's theorem

In mathematics, Lusin's theorem (more properly Luzin's theorem, named for Nikolai Luzin) in real analysis is another form of Littlewood's second principle.

It states that every measurable function is almost a continuous function:

For an interval $[a,b]$, let $f:[a,b]\rightarrow \mathbb {C}$ be a measurable function. Then $\forall \epsilon >0$, there exists a compact $E\subset [a,b]$ such that f restricted to E is continuous and $\mu (E^{C})<\epsilon$. $E^{C}$ denotes the complement of E. Notice E inherits a subspace topology from $[a,b]$, and it is in this topology we define continuity of f restricted to E.

A simple proof is as follows. Recall the continuous functions are dense in $L^{1}[a,b]$. Therefore there exists a sequence of continuous functions $\{g_{n}\}$ s.t. $\{g_{n}\}\rightarrow f$ in $L^{1}$. From this sequence, we can extract a subsequence $\{g_{n_{k}}\}$ such that $g_{n_{k}}\rightarrow f$ almost everywhere. By Egorov's theorem, we have $g_{n_{k}}\rightarrow f$ uniformly except on some set of arbitrarily small measure. Since continuous functions are closed under uniform convergence, the theorem is proved.