I'm trying to prove the following statements:
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space. Let $f_1, f_2, … \in \mathcal{L}^1(\mu)$ be nonnegative and such that $\lim_{n \to \infty} \int f_n d\mu$ exists. Assume $f_n \to f$ $\mu$-almost everywhere, where $f$ is measurable.
(i) Show that $f \in \mathcal{L}^1{(\mu)}$ and
(ii) $\lim_{n \to \infty} \int |f_n – f|d\mu = \lim_{n \to \infty} \int f_n d\mu – \int f d\mu$.
The first statement I managed to prove using Fatou's lemma. Namely, we have
$$
\int f d\mu = \int \lim_{n \to \infty}d\mu \leq \lim_{n \to \infty} \int f_n d\mu < \infty,
$$
and so $f \in \mathcal{L}^1{(\mu)}$. For the second statement, we know by the linearity of the integral that $\lim_{n \to \infty} \int f_n d\mu – \int f d\mu = \lim_{n \to \infty} \int (f_n – f) d\mu$. Here I think I can somehow invoke the idea that $f_n$ is close to $f$ for large $n$ except for a $\mu$-null set, but I am not sure how to continue with this. Any help is greatly appreciated, thanks! For those who want to look it up, this is exercise 4.2.2 in Achim Klenke's Probability Theory.
Best Answer
Second part is essentialy Scheffe's Lemma. $\int (f-f_n)^{+} \to 0$ by DCT with dominating function $f$.
Let $L=\lim \int f_n$. Then $\int (f-f_n) \to \int f -L$. Now $\int (f-f_n)^{-} =\int (f-f_n)^{+} -\int (f-f_n) \to L-\int f$. So $\int |f-f_n| =\int (f-f_n)^{+}+\int (f-f_n)^{-} \to L-\int f$.