one interesting qn related to convergence of integration.. let $f_n:\mathbb R\to \mathbb R$ be a nonnegative sequence of Borel measurable functions which converge to a function $f,$ and the integral of all $f_n$ and of $f$ are all $1.$ Then does the integral of $|f_n-f|$ goes to 0 as $n \to\infty?$
any help will be appreciated.
 A: General Lebesgue Dominated Convergence Theorem:-
Let $ \{g_{n}\}$ be a sequence of integrable function which converge almost surely to an integrable function $g$. If $\{f_{n}\}$ be a sequence of functions such that $|f_{n}|\leq g_{n}$ and $\{f_{n}\}$ converges to $f$ almost surely. If $\int_{X}g\,d\mu=\lim_{n\to\infty}\int_{X}g_{n}\,d\mu$ then $\int_{X}f\,d\mu=\lim_{n\to\infty}\int_{X}f_{n}\,d\mu$
More generally:-
If $\{f_{n}\}$ is a sequence of integrable functions from a measure space $(X,\mathcal{F},\mu$) and $f_{n}\to f$ almost surely and $f$ is integrable. Then $\int_{X}|f_{n}-f|\,d\mu\to 0$ iff $\lim_{n\to\infty}\int_{X}|f_{n}|\,d\mu=\int_{X}|f|\,d\mu$.
$|f_{n}-f|\leq |f_{n}|+|f|\,\,,(=g_{n})$(as in the theorem)
and $\lim_{n\to\infty}\int_{\mathbb{R}}(|f_{n}|+|f|)\,d\lambda=2\int_{\mathbb{R}}|f|\,d\lambda$.
(So $\int_{X}g\,d\mu=\lim_{n\to\infty}\int_{X}g_{n}\,d\mu$)
And each $f_{n}$ and $f$ are integrable by assumption .
So applying Dominated Convergence Theorem you get that $\lim_{n\to\infty}\int_{\mathbb{R}} |f_{n}-f|\,d\lambda=\int\lim_{n\to\infty}|f_{n}-f|\,d\lambda= \int_{\mathbb{R}}0\,d\lambda=0$.
