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?
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.