ask a question about generalized Dominated Convergence Theorem

Let $(X,\mathfrak{M},\mu)$ be a measure space. Suppose $f_n, f \in L^1 (\mu)$ and $f_n \to f$ pointwise. Then

$$\lim \limits_{n \to \infty} \int_X |f_n -f| d\mu =0 \iff \lim \limits_{n \to \infty} \int_X |f_n| d\mu = \int_X |f| d\mu$$

Proof:

“$\Longleftarrow$" part: From $$\lim \limits_{n \to \infty} \int_X |f_n| d\mu = \int_X |f| d\mu ,$$

we can get $$\lim \limits_{n \to \infty} \int_X |f_n| d\mu - \int_X |f| d\mu =0$$

$$\lim \limits_{n \to \infty} \int_X (|f_n| - |f|) d\mu =0$$

Now I am stuck. Notice that the LHS of the original question is $|f_n -f|$. What I get is $|f_n| -|f|$. They are different, and I cannot get my desired result.

"$\Longrightarrow$" part: In this direction, we have $\lim \limits_{n \to \infty} \int_X |f_n -f| d\mu =0$, which implies $\lim \limits_{n \to \infty} \int_X f_n d\mu = \int_X f d\mu$ .

This is equivalent to saying $\int_X f_n d\mu \to \int_X f d\mu$. If I can show $|f_n -f| \leq f_n$, then by the generalized Dominated Convergence Theorem, I can immediately get $\int_X |f_n-f| d\mu \to \int_X f d\mu$, which would complete the proof. But I don't know whether I can show $|f_n -f| \leq f_n$.

Solution 1:

($\Rightarrow$) Assume that $\int|f_n-f|\to0$. We have that $|f_n-f|\geq ||f_n|-|f||$. Let $N_1$ and $N_2$ be the sets of $n$ where $|f_n|\geq |f|$ and $|f_n|<|f|$ respectively. For $n\in N_1$ we have $\int|f_n-f|\geq \int|f_n|-\int|f|\geq0$. Therefore, the limit $\int|f_n|-\int|f|$ along $n\in N_1$ is $0$. Similarly, for $n\in N_2$, $\int|f_n-f|\geq \int|f|-\int|f_n|\geq0$. Therefore, the limit along $n\in N_2$ is also $0$. Hence $\int|f_n|\to\int|f|$.

($\Leftarrow$) Assume now that $\int|f_n|\to\int|f|$. The function $|f_n-f|$ converges pointwise a.e. to $0$ and it satisfied $|f_n-f|\leq |f_n|+|f|$. The functions $g_n=|f_n|+|f|\in L^1(\mu)$ and converge pointwise to $|f|+|f|$. Moreover, $\int g_n=\int (|f_n|+|f|)=\int|f_n|+\int|f|$ converges to $\int (|f|+|f|)$, by the assumption. Therefore, by the DCT we have that $\int|f_n-f|\to \int 0$.