Lusin's theorem

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

It states that every measurable function is a continuous function on nearly all its domain:

For an interval ["a", "b"] , let

:$f:\; [a,b]\; ightarrow\; mathbb\{C\}$

be a measurable function. Then given $scriptstyle\; varepsilon\; >\; 0$, there exists a compact $scriptstyle\; E\; subset\; [a,b]$ such that ƒ restricted to "E" is continuous and

:

Here "E"^"c" denotes the complement of "E". Note that "E" inherits the subspace topology from ["a", "b"] ; continuity of ƒ restricted to "E" is defined using this topology.

A proof of Lusin's theorem

Since ƒ is measurable, it is bounded on the complement of some open set of arbitrarily small measure. So, redefining ƒ to be 0 on this open set if necessary, we may assume that ƒ is bounded and hence integrable. Since continuous functions are dense in L¹ ["a","b"] , there exists a sequence of continuous functions "g"_"n" tending to ƒ in the L¹ norm. Passing to a subsequence if necessary, we may also assume that "g"_"n" tends to ƒ almost everywhere. By Egorov's theorem, it follows that "g"_"n" tends to ƒ "uniformly" off some open set of arbitrarily small measure. Since uniform limits of continuous functions are continuous, the theorem is proved.

References

* N. Lusin. Sur les propriétés des fonctions mesurables, Comptes Rendus Acad. Sci. Paris 154 (1912), 1688-1690.