I came across the following problem in my self-study:
If $f_n, f$ are integrable and $f_n \rightarrow f$ a.e. on $E$, then $\int_E |f_n – f| \rightarrow 0$ iff $\int_E |f_n| \rightarrow \int_E |f|$.
I am trying to prove (1) and the book I am using suggests that it follows from the Generalized Lebesgue Dominated Convergence Theorem:
Let $\{f_n\}_{n=1}^\infty$ be a sequence of measurable functions on $E$ that converge pointwise a.e. on $E$ to $f$. Suppose there is a sequence $\{g_n\}$ of integrable functions on $E$ that converge pointwise a.e. on $E$ to $g$ such that $|f_n| \leq g_n$ for all $n \in \mathbb{N}$. If $\lim\limits_{n \rightarrow \infty}$ $\int_E$ $g_n$ = $\int_E$ $g$, then $\lim\limits_{n \rightarrow \infty}$ $\int_E$ $f_n$ = $\int_E$ $f$.
I suspect that I need the right inequalities to help satisfy the hypothesis of the GLDCT, but I am not certain about what these inequalities should be.
Best Answer
The fact that $$\int_E |f_n - f| \to 0 \Rightarrow \int_E |f_n| \to \int_E |f|$$ is actually a simple consequence of the fact that $f \mapsto \int_E |f|$ is a norm; in other words, it holds independently of the assumptions given. More specifically, in any normed space, the following inequality holds $$ |\| x \| - \| y \| | \leq \| x - y \|.$$ The proof is easy: if $\| x \| \geq \| y \|$, write $\| x \| = \| x - y + y \| \leq \| x - y \| + \| y \|$, so that $0 \leq \|x\| - \| y \| \leq \| x - y \|$. If $\| y \| > \| x \|$, reverse the roles of $x$ and $y$, to get that $\| y \| - \| x \| \leq \| x -y \|$. Taking these inequalities together proves the result.
Hence $\| f_n - f\|_1 \geq |\| f_n\|_1 - \| f \|_1|$, and since the left hand side tends to $0$, so must the right hand side, since it is $\geq 0$. This proves the first implication.
For the converse, assume that $$ \int_E |f_n | \to \int_E |f|$$ and that $f_n \to f$ a.e. on $E$. We can, as Elan B. mentions, take $g_n = |f_n| + |f|$ and conclude using the triangle inequality that $|f_n - f| \leq g_n$. By assumption, we have $|f_n| \to |f|$ a.e. on $E$ (since, again, $|f_n - f | \to 0$ and $|f_n - f| \geq | |f_n| - |f| |$). Furthermore, we have $$ \int_E g_n = \int_E (|f_n| + |f|) \to \int_E 2|f|.$$ From the generalized DCT, we conclude that $$\int_E |f_n - f| \to 0,$$ as claimed. This proves (1).