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{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 μ(EC) < ε. EC 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 L1[a,b]. Therefore there exists a sequence of continuous functions {gn} s.t. \{ g_n\} \rightarrow f in L1. 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.

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