Suppose $\displaystyle f$ is a non-negative integrable function on $\displaystyle X$.For each $\displaystyle n$,define $\displaystyle E_n= \{ x \in X : f(x)>n \}$.

Show that $\displaystyle lim_{n \rightarrow \infty} m(E_n) =0$.

($\displaystyle m(A)$ is the measure of $\displaystyle A$.)

I was trying to prove that since $\displaystyle f$ is integrable,then $\displaystyle f < \infty a.e.$.Since $\displaystyle E_n$ is a decreasing sequence of sets,so $\displaystyle lim_{n \rightarrow \infty} m(E_n) = m( \bigcap_{n \in N} E_n)$

Since $\displaystyle f< \infty a.e$ so $\displaystyle \bigcap_{n \in N} E_n = \phi$

so $\displaystyle m( \phi ) = 0 $ and hence get the result.

but the problem is I have to assume $\displaystyle m(E_1) < \infty$ to claim the result.I was stuck here.

Can anyone help?Or if this is not the correct approach,can anyone show me the right way?