Proposition 7.4.15: Measurable Functions are Almost Continuous
Suppose f is a measurable function defined on an interval [a, b] such that the set where f is plus or minus infinity has measure zero. Then, for any > 0 we can find a step function g and a continuous function h such that
| f(x) - g(x) | <except on a set of measure less than .
| f(x) - h(x) | <