Lusin's theorem: Difference between revisions

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 function is nearly continuous".

For an interval [a, b], let

${\displaystyle f:[a,b]\rightarrow \mathbb {C} }$

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

${\displaystyle \mu (E)>b-a-\varepsilon .\,}$

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

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

${\displaystyle f:X\rightarrow Y}$

be a measurable function. Given ε > 0, for every ${\displaystyle A\in \Sigma }$ 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 ${\displaystyle f_{\varepsilon }:X\rightarrow Y}$ 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.

Since f is measurable, it is bounded on the complement of some open set of arbitrarily small measure. So, redefining f to be 0 on this open set if necessary, we may assume that f 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 f in the L¹ norm. Passing to a subsequence if necessary, we may also assume that g_n tends to f almost everywhere. By Egorov's theorem, it follows that g_n tends to f uniformly off some open set of arbitrarily small measure. Since uniform limits of continuous functions are continuous, the theorem is proved.

• 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 3

• W. Zygmunt. Scorza-Dragoni property (in Polish), UMCS, Lublin, 1990