Luzin's theorem

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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] 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 [ \ ] s.t. [ \ \rightarrow f] in [L^1]. From this sequence, we can extract a subsequence, also called [ \ ] as a slight abuse of language, s.t. [ g_n \rightarrow f] almost everywhere. By Egorov's theorem, we have [ g_n \rightarrow f] uniformly except on some set of arbitrarily small measure. Since continuous functions are closed under uniform convergence, the theorem is proved.

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