Lusin's theorem
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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]\rightarrow \mathbb{C}](../../../../math/7/b/9/7b94a3de27138fba7d5b93cde31d875c.png)
be a measurable function. Then given 
, there exists a compact ![\scriptstyle E\ \subset\ [a,b]](../../../../math/0/3/e/03e0d936c4b3594227130f4540ef537d.png)
such that ƒ restricted to E is continuous and
Here Ec 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.
[edit] 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 gn tending to ƒ in the L1 norm. Passing to a subsequence if necessary, we may also assume that gn tends to ƒ almost everywhere. By Egorov's theorem, it follows that gn tends to ƒ uniformly off some open set of arbitrarily small measure. Since uniform limits of continuous functions are continuous, the theorem is proved.
[edit] References
- N. Lusin. Sur les propriétés des fonctions mesurables, Comptes Rendus Acad. Sci. Paris 154 (1912), 1688-1690.
