In the mathematical field of real analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) states that every measurable function is a continuous function on nearly all its domain. In the informal formulation of J. E. Littlewood, "every measurable function is nearly continuous".
For an interval [a, b], let
be a measurable function. Then, for every ε > 0, there exists a compact E ⊂ [a, b] such that f restricted to E is continuous and
Note that E inherits the subspace topology from [a, b]; continuity of f restricted to E is defined using this topology.
be a measurable function. Given ε > 0, for every of finite measure there is a closed set E with µ(A \ E) < ε such that f restricted to E is continuous. If A is locally compact, we can choose E to be compact and even find a continuous function with compact support that coincides with f on E.
Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain.
A proof of Lusin's theorem
Since f is measurable, there exists a sequence of step functions, fn converging to f pointwise almost everywhere. Each fn is bounded on a set of finite measure, hence integrable. By Egorov's theorem, may take a closed set E, such that the measure of A \ E is arbitrarily small, and such that fn converges to f uniformly. Thus f is in L1(A). Since continuous functions are dense in L1, we may approximate f with a continuous function defined on A.
- N. Lusin. Sur les propriétés des fonctions mesurables, Comptes Rendus Acad. Sci. Paris 154 (1912), 1688–1690.
- G. Folland. Real Analysis: Modern Techniques and Their Applications, 2nd ed. Chapter 2
- W. Zygmunt. Scorza-Dragoni property (in Polish), UMCS, Lublin, 1990