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".

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 and

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

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

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

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