Royden's Real Analysis Question: Let {$f_n$} be a sequence of nonnegative measurable functions on $R$ such that $f_n\implies f$ pointwise on $E$. Let $M\geq0$ be such that $\int_Ef_n\leq M$ for all $n$. I want to show that $\int_Ef\leq M$. Also I want to verify that this property is equivalent to the statement of Fatou's Lemma.
Approach: Here is how I started.. Since $f_n\implies f$ on $E$ then by Fatou's Lemma $\int_Ef\leq lim\inf\int_Ef_n$ where $inf\int_Ef_n\leq M$ this implies $$\int_Ef\leq M$$ Hence verifying Fatou"s Lemma as well.
I know I am wrong somewhere but I will appreciate any help given.
Thankyou!
What you showed is that Fatou's lemma implies the mentioned property. Now you have to show that this property implies Fatou's lemma.
Let $(f_n,n\in\Bbb N)$ be a sequence of measurable integrable functions and $a_N:=\inf_{k\geqslant N}\int f_kd\mu$. Working with the sequence $(f_n,n\geqslant N)$ (for which the sequence of integrals has the same $\liminf$ as those of the whole sequence), one can see that that $\int fd\mu\leqslant a_N$ for each $N$. Now take the limit $\lim_{N\to +\infty}$.