Let $(\Omega,\mathcal{F},P)$ be a probability space, and $A\in\mathcal{F}$ such that $P(A)>0$. Let $\sigma(A)=\{\emptyset,A,\Omega\backslash A,\Omega\}$ denote the sigma filed generated by $A$. Show that for any $\omega\in A$ and r.v. $X\in L^1$
$$E[X|\sigma(A)](\omega)=\frac{1}{P(A)}\int_\Omega X1_A(\omega)dP=\frac{1}{P(A)}\int_AXdP$$
This seems quite obvious but am quite stuck. I started with the definition of conditional expectation but am getting nowhere, the only thing I can see is, $E[X|\sigma(A)]=E[X|\sigma(A)]1_A+E[X|\sigma(A)]1_{\Omega\backslash A}$, and can't see where I can pull out $1/P(A)$, so any help will be greatly appreciated. Thanks