Lusin's Theorem

Let $f(x)$ be a finite and Measurable Function in $(-\infty,\infty)$, and let $\epsilon$ be freely chosen. Then there is a function $g(x)$ such that

1. $g(x)$ is continuous in $(-\infty,\infty)$,

2. The Measure of $\{x : f(x)\not=g(x)\}$ is $<\epsilon$,

3. $M(\vert g\vert; R_1)\leq M(\vert f\vert; R_1)$,

where $M(f;S)$ denotes the upper bound of the aggregate of the values of $f(P)$ as $P$ runs through all values of $S$.

References

Kestelman, H. §4.4 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 30 and 109-112, 1960.