Carathéodory's theorem (conformal mapping)

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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".

Classical statement

For an interval [a, b], let

$f:[a,b]\to \mathbb{ℂ}$

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

$\mu (E)>b-a-\epsilon .$

Note that E inherits the subspace topology from [a, b]; continuity of f restricted to E is defined using this topology.

General form

Let $(X,\mathrm{\Sigma },\mu )$ be a Radon measure space and Y be a second-countable topological space, let

$f:X\to Y$

be a measurable function. Given ε > 0, for every $A\in \mathrm{\Sigma }$ 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 $f_{\epsilon }:X\to Y$ 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, f_n converging to f pointwise almost everywhere. Each f_n 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 f_n 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.

References

• 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