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 almost everywhere 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 , 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.
- 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