Any body can help me to prove this problem in simple function?
Let if $g_n \geq 0$, $g_n \rightarrow g$ and $\int g_n d\mu \leq E \leq \infty$ then $\int g_n d\mu \leq E$ with $(S, \Sigma, \mu)$ as measurable function and $E\in \Sigma$.
Hope that this is will be clear, thanks..
By Fatou's Lemma we have that since $g_{n}\ge0$ and $g_{n}\to g$ then
$\int gd\mu\le\lim\inf_{n\to\infty}\int g_{n}d\mu\le E$.
(I have assumed that you wanted to prove $\int gd\mu\le E$ since otherwise it follows directly from the statement of the question. Also I think you meant $E\in[0,\infty)$)