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[]

Sources

• 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

Citations