# Tag Archives: Measure

## f > 0 implies int f(x) dx > 0

If $$f : \mathbb{R} \to \mathbb{R}$$ is continuous on $$[a,~b]$$ and $$f > 0$$ then $\int_{a}^{b} f(x) ~ dx > 0 .$ This theorem is a well-known example in undergraduate analysis and can be proven easily. But what if the continuity of $$f$$ is replaced by integrability? Theorem. If \(f : \mathbb{R} \to \mathbb{R}… Read More »