Variants of Fatou's Lemma

Question

Fatou's Lemma is

Let $(\Omega, \Sigma, \mu)$ be a measure space and $X \in \Sigma$. Let $f_n; f_n : X \to [0, +\infty]$ be a sequence of $(\Sigma, \mathcal{B}_{\mathbb{R}_{\geq 0}})$-measurable functions. Define $f(x) = \lim\inf_{n\to\infty} f_n$. Then $f$ is $(\Sigma, \mathcal{B}_{\mathbb{R}_{\geq 0}})$-measurable and

$\begin{align} \int_X f d\mu \leq \liminf_{n\to\infty} \int_X f_n d\mu \end{align}$

• Is this true if you generalize the codomain of $f$ and $f_n$ to be all of $\mathbb{R}$ and $(\Sigma, \mathcal{B}_{\mathbb{R}})$-measurable?

• Is this true if you replace $\liminf$ with $\limsup$?

Provide counterexamples please if possible. No this is not a homework question.

Answer 1

The answers to both questions are no. Here's a counterexample for both cases.

Let $X = \mathbb{R}$. Let $f_{n}(x)=-1$ if $x \in [n, n+1)$, and $0$ otherwise. Then $$ 0=\int f d\mu > \liminf_{n} \int_{n} f_{n} d\mu = \limsup_{n} \int f_{n} d\mu =-1. $$

Answer 2

If $f_n$ is a sequence of measurable functions bounded above by an integrable function $f$, i. e., $f_n\leq f$ with $f\in L_1$, then $$\limsup_n\int f_n\leq \int\limsup_n f$$

Suppose $f_n\leq A$ for all $n$. Then $f-f_n$ is a sequence of nonnegative measurable functions. The classical Fatou's lemma implies \begin{align} \int f\,d\mu+\int\liminf_n(-f_n)\,d\mu&=\int\liminf_n(f-f_n)\,d\mu\\ &\leq \liminf_n\int f-f_n\,d\mu=\int f+\liminf_n(-\int f_n\,d\mu) \end{align}