# The finite integral

Suppose that $f \in L^1([0,1])$, and $h(t)$ denotes the Lebesgue measure of the set $\lbrace x \in [0,1] \; ; \; |f(x)|> t \rbrace$ for $0<t<\infty$.Show that $$\int_0^{\infty} h(t)dm(t) < \infty$$.

I have a qualifying exam in coming January and I have tried to solve this question but I was not success. I just use the fact since $f \in L^1([0,1])$, thus $h(t)<\infty$.

This is just an application of Fubini's theorem \begin{align*} \int_0^\infty h(t) \, dt &= \int_0^\infty \int_\mathbb{R} I\{|f(x)| > t\} \, dx \, dt = \int_\mathbb{R} \int_0^\infty I\{|f(x)| > t\} \, dt \, dx \\ &= \int_\mathbb{R} \int_0^{|f(x)|} 1 \, dt \, dx = \int_\mathbb{R} |f(x)| \, dx < \infty \end{align*}