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'. 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 , let be a measurable function. Then, for every ε > 0, there exists a compact E ⊂  such that f restricted to E is continuous almost everywhere and Note that E inherits the subspace topology from ; continuity of f restricted to E is defined using this topology. Let ( X , Σ , μ ) {displaystyle (X,Sigma ,mu )} 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 ε > 0 {displaystyle varepsilon >0} , for every A ∈ Σ {displaystyle Ain Sigma } of finite measure there is a closed set E {displaystyle E} with μ ( A ∖ E ) < ε {displaystyle mu (Asetminus E)<varepsilon } such that f {displaystyle f} restricted to E {displaystyle E} is continuous. If A {displaystyle A} is locally compact, we can choose E {displaystyle E} to be compact and even find a continuous function f ε : X → Y {displaystyle f_{varepsilon }:X ightarrow Y} with compact support that coincides with f {displaystyle f} on E {displaystyle E} and such that   sup x ∈ X | f ε ( x ) | ≤ sup x ∈ X | f ( x ) | {displaystyle sup _{xin X}|f_{varepsilon }(x)|leq sup _{xin X}|f(x)|} .