# Variants of Fatou's Lemma

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.

• For second question, search for reverse Fatou's lemma
– user608030
Oct 25, 2018 at 4:53
• Zachary Selk your answer was extremely useful. Thank you. Oct 25, 2018 at 4:55
• The first question is equivalent to the second. Just replace the functions in the counterexample to second question by their negatives. Oct 25, 2018 at 5:34

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