A generalization of Lusin’s theorem

Abstract:

In this note we characterize $\sigma$-finite Riesz measures that allow one to approximate measurable functions by continuous functions in the sense of Lusin’s theorem. We call such measures Lusin measures and show that not all $\sigma$-finite measures are Lusin measures. It is shown that if a topological space $X$ is either normal or countably paracompact, then every measure on $X$ is a Lusin measure. A counterexample is given to show that these sufficient conditions are not necessary.