Question

If f is a Lebesgue integrable function on a measureable set E of finite measure and En={x∈E:∣f(x)∣≥n}, then limn→∞(n⋅m(En))=0, where m(En) denotes the measure of En.

Answer 1

Hint
\[

\int_E |f| dm \ge \int_{E_n} |f| dm \ge n\, m(E_n)
\]

Answer 2

\[
m(E_n) \le \frac 1 n \int_E |f| dm
\]
when n goes to infinity, then the right hand side goes to zero

Answer 3

thanks!