Question on a measurable function

Question

Suppose $f$ is a nonnegative measurable function on $\mathbb{R}$. We assume that $f$ can take infinite values . The Lebesgue integral of such a function is defined exactly as for functions from the class $K_3$. We also assume that $ \int f dx <\infty $ . Prove that
$$m\{x\in \mathbb{R} : f(x)=\infty \}=0$$. any hint and help will be appreciated .

Answer 1

Suppose $f\ge 0$, $f$ is measurable and that $\int f < \infty$. I will use the notation $|\cdot|$ for Lebesgue measure and $[f > \lambda]$ for $f^{-1}(\lambda,\infty)$.

Then by Chebyshev's inequality, $$ |[f > \lambda]|\le {1\over \lambda} \int f.$$ Since the right-hand side converges to zero as $\lambda\to\infty$, you have $$|[f = \infty] = \left|\bigcap_{\lambda > 0} [f > \lambda]\right| = 0.$$