Lusin's theorem

In the mathematical field of real analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) or Lusin's Criterion states that an almost-everywhere finite function is measurable if and only if it 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

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

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.

Also for any function f, defined on the interval [a, b] and almost-everywhere finite, if for any ε > 0 there is a function ϕ, continuous on [a, b], such that the measure of the set

${\displaystyle \{x\in [a,b]:f(x)\neq \phi (x)\}}$

is less than ε, then f is measurable.[1]

General form

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

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

be a measurable function. Given ${\displaystyle \varepsilon >0}$, for every ${\displaystyle A\in \Sigma }$ of finite measure there is a closed set ${\displaystyle E}$ with ${\displaystyle \mu (A\setminus E)<\varepsilon }$ such that ${\displaystyle f}$ restricted to ${\displaystyle E}$ is continuous. If ${\displaystyle A}$ is locally compact, we can choose ${\displaystyle E}$ to be compact and even find a continuous function ${\displaystyle f_{\varepsilon }:X\rightarrow Y}$ with compact support that coincides with ${\displaystyle f}$ on ${\displaystyle E}$ 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.

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
• Lawrence C. Evans, Ronald F. Gariepy, "Measure Theory and fine properties of functions", CRC Press Taylor & Francis Group, Textbooks in mathematics, Theorem 1.14

1. https://encyclopediaofmath.org/wiki/Luzin_criterion