I was going through the proof of the Dominated Convergence Theorem.
Now if we have that ($f$$_n$) is a sequence of measurable functions such that $\lvert f_n\rvert$ $\le$ $g$ for all n where g is integrable on $\Bbb{R}$.
And if $f$ = $\lim_{n}$$f_n$ almost everwhere.
We can show that ($g+f_n$) is a sequence of non-negative measurable functions.
Then by Fatou's lemma, we have that $\int$liminf($g+f_n$)$d$$x$ $\le$ liminf$\int$($g+f_n$)$d$$x$.
Now from here, we can obtain that
$\int$($g+f$)$d$$x$ $\le$ $\int$$gdx$+liminf $\int$$f_ndx$.How?
Please explain this last step.I know that since both are integrable, the integral can be seperated..but how is liminf seperated in the right-hand side?
Best Answer
Inside the liminf, $\int g$ is just a number. So what it says is that $$ \liminf_n(c+f_n)=c+\liminf _nf_n, $$ where $c=\int g$.