Lusin's theorem

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

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

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

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

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

General form

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

$f:X\rightarrow Y$

be a measurable function. Given ε > 0, for every $A\in \Sigma$ 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 $f_{\varepsilon }:X\rightarrow Y$ with compact support that coincides with f on E and such that ${\displaystyle \ \sup _{x\in X}|f_{\varepsilon }(x)|\leq \sup _{x\in X}|f(x)|}$.

Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain.

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
• M. B. Feldman, "A Proof of Lusin's Theorem", American Math. Monthly, 88 (1981), 191-2