## 7.4. Lebesgue Integral

### 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 thatexcept on a set of measure less than .| f(x) - g(x) | <

| f(x) - h(x) | <