kreamykraka80

2022-07-10

Let $\left({f}_{n}{\right)}_{n\in \mathbb{N}}$ be a sequence of elements in $M\left(X,S\right)$, let $g\in {M}^{+}\left(X,S\right)$ such that ∫gdμ<∞ and in $E\in S$. Then, it follows from Fatou's lemma that:
${\int }_{E}\underset{n\to \mathrm{\infty }}{lim inf}\left({f}_{n}\right)d\mu \le \underset{n\to \mathrm{\infty }}{lim inf}{\int }_{E}{f}_{n}d\mu$
Question 1 Can someone please give me a reference for the above generalised fatou's lemma.
Question 2 What is the condition on $\left({f}_{n}{\right)}_{n\in \mathbb{N}}$ so that we may drop liminf and we have
${\int }_{E}\underset{n\to \mathrm{\infty }}{lim}\left({f}_{n}\right)d\mu \le \underset{n\to \mathrm{\infty }}{lim}{\int }_{E}{f}_{n}d\mu$
My try: Define
${h}_{n}={f}_{n}+g$
Then ${h}_{n}\ge 0$. Applying fatou's lemma to ${h}_{n}$ we have
${\int }_{E}\underset{n\to \mathrm{\infty }}{lim inf}\left({h}_{n}\right)d\mu \le \underset{n\to \mathrm{\infty }}{lim inf}{\int }_{E}{h}_{n}d\mu$
So we get
${\int }_{E}\underset{n\to \mathrm{\infty }}{lim inf}\left({f}_{n}+g\right)d\mu \le \underset{n\to \mathrm{\infty }}{lim inf}{\int }_{E}\left({f}_{n}+g\right)d\mu$
How do we conclude?

Perman7z

Expert

Let ${g}_{n}\left(x\right)=inf\left\{{f}_{i}\left(x\right):i\ge n\right\}$. Then ${g}_{n}\left(x\right)$ increases monotonically to $lim inf{f}_{n}$ and by the monotone convergence theorem, $\int {g}_{n}=\int lim inf{f}_{n}$. Since ${g}_{n}\le {f}_{n}$ for every $n$, we have $\int {g}_{n}\le \int {f}_{n}$. Your result then follows.