19.6.1.1 Lebesgue measure and integration

Problem: Let \( \mu(A)<\infty \). Prove that the non-negative function \( f \) is integrable with respect to \( A \) only when the following series converges: \[ \sum_{n=1}^{\infty} 2^{n} \mu\left(A \cap\left\{x: f(x) \geq 2^{n}\right\}\right) . \]