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