# Fatou's lemma extension and dropping lim inf.

Let $$(f_n)_{n\in \Bbb N}$$ be a sequence of elements in $$M(X,S)$$, let $$g\in M^+(X,S)$$ such that $$\int gd\mu<\infty$$ and $$f_n\ge-g\$$ $$(a.e.- \mu)\ \forall n\in \Bbb N\$$ in $$E\in S$$. Then, it follows from Fatou's lemma that:

$$\int_{E} \liminf_{n\to \infty} (f_n)d\mu\le \liminf_{n\to \infty} \int_E f_nd\mu$$ Question 1 Can someone please give me a reference for the above generalised fatou's lemma.

Question 2 What is the condition on $$(f_n)_{n\in \Bbb N}$$ so that we may drop $$\lim \inf$$ and we have $$\int_{E} \lim_{n\to \infty} (f_n)d\mu\le \lim_{n\to \infty} \int_E f_nd\mu$$ My try: Define $$h_n=f_n+g$$ Then $$h_n\geq 0$$. Applying fatou's lemma to $$h_n$$ we have $$\int_{E} \liminf_{n\to \infty} (h_n)d\mu\le \liminf_{n\to \infty} \int_E h_nd\mu$$ So we get $$\int_{E} \liminf_{n\to \infty} (f_n+g)d\mu\le \liminf_{n\to \infty} \int_E (f_n+g)d\mu$$ How do we conclude?

• I don't have the book on me to confirm, but this should probably be in Real Analysis by Royden and Fitzpatrick?
– Zim
Commented Dec 2, 2021 at 4:35
• @Zim So both of my questions are answered there? Can you please give a brief idea for the proof.
– user999627
Commented Dec 2, 2021 at 4:46
• This is just Fatou's Lemma applied to $f_n+g$. Nobody would state this separetely as a theorem. Commented Dec 2, 2021 at 5:05
• @Kavi Rama Murthy So the condition that $g>0$ is needed or not?
– user999627
Commented Dec 2, 2021 at 5:10
• No. It is not needed. Commented Dec 2, 2021 at 5:11

Let $$g_n(x)=\inf\{f_i(x) : i \geq n \}$$. Then $$g_n(x)$$ increases monotonically to $$\liminf f_n$$ and by the monotone convergence theorem, $$\int g_n = \int \liminf f_n$$. Since $$g_n \leq f_n$$ for every $$n$$, we have $$\int g_n \leq \int f_n$$. Your result then follows.