Suppose there are constants $K > 0$ and $p > 1$ such that $μ\{x∈X : |f(x)|>M\}< \frac{K}{M^p}$ for all $M >0.$

Question

Let $f$ be a measurable function on a measure space $(X,μ),$ where $μ$ is a finite measure. Suppose there are constants $K > 0$ and $p > 1$ such that $μ\{x∈X : |f(x)|>M\}< \frac{K}{M^p}$ for all $M >0.$ Prove that f is integrable.

This is a past analysis qual problem that I seem to be having some great difficulty with. Some help would be awesome. Thanks.

Answer 1

Hint Use Tonelli's theorem to deduce that $$\int_X |f(x)| \, d\mu(x) = \int_X \int_0^{|f(x)|} \, dr \, d\mu(x) = \int_0^{\infty} \mu(\{x \in X; |f(x)|>r\}) \, dr.$$