# 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

be a measurable function. Then, for every *ε* > 0, there exists a compact *E* ⊆ [*a*, *b*] such that *f* restricted to *E* is continuous almost everywhere and

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

## General form

Let be a Radon measure space and *Y* be a second-countable topological space equipped with a Borel algebra, and let

be a measurable function. Given , for every of finite measure there is a closed set with such that restricted to is continuous. If is locally compact, we can choose to be compact and even find a continuous function with compact support that coincides with on and such that .

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

## On the proof

The proof of Lusin's theorem can be found in many classical books. Intuitively, one expects it as a consequence of Egorov's theorem and density of smooth functions. Egorov's theorem states that pointwise convergence is nearly uniform, and uniform convergence preserves continuity.

## References

- N. Lusin. Sur les propriétés des fonctions mesurables,
*Comptes rendus de l'Académie des Sciences de Paris*154 (1912), 1688–1690. - G. Folland.
*Real Analysis: Modern Techniques and Their Applications*, 2nd ed. Chapter 7 - 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