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}

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

\mu ( E^c ) < \varepsilon.

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.
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