Implications with the Lebesgue integral

Question

Let $(X,\mathcal{B},\mu)$ a probability space and $f:X \to \mathbb{R}$ a non negative and integrable function. My question is, for $a>0$,

if $\int_{X}fd\mu\geq a$, then there exist a Borel set, $B\in \mathcal{B}$, with $\mu(B)>0$ such that $f(x)\geq a$, for all $x\in B$ ?

Please, help me! Thanks.

Answer 1

Yes. Take $B=\{x:f(x) \geq a\}$. If $\mu (B)=0$ the $f<a $ almost everywhere which implies $\int f \, d\mu <a$.