I have question ,let $f$ is measurable and integrable i.e $f\in(\Omega,\mathbf{A}, \mathbb{R})$ and for all $A\in \mathbf{A}$ $\int\_Af(\omega)d\mu=0$ show that $f=0$ almost every where.

answer: I wil solve it from couter example sps that f is not 0 ie $f(\omega)>0$so I define the set ${\omega \in \Omega :f(\omega)>0}$ and I know that the set ${ \omega \in\Omega :f(\omega)>\epsilon}$ is a subset of $f(\omega)>0$ and from hypothesis of question I know that $\int f(\omega)>0=0$ so the $\int f(\omega)<\epsilon$ is also 0 how I can cuncluderen that $f(\omega)=0$ almost every where