 Comparison test for integrals

If measurable functions satisfy $|f|\le g$ almost everywhere and $g$ is nonnegative and Lebesgue integrable, then $f$ is Lebesgue integrable and $\int |f|\le \int g$. This follows from monotonicity of the Lebesgue integral. Conversely, $0\le g\le |f|$ with $\int g=\infty$ rules out Lebesgue integrability of $f$. For nonnegative improper integrals, the same comparison applies after taking increasing finite-interval limits.