a homework question from measure and integraiton theory course.
Suppose $f_n \in L_1(\mathbb R^d)$ for each $n\in\mathbb N$， $f_n\geq 0$ ， $f_n\to f$ a.e and $\int f_n\to\int f<\infty$.
Prove that $\int|f_n-f| \to 0$(Hint: ($f_n – f)_-\leq f$. Use dominated convergence theorem )
I am thinking that since $|f_n -f | \to 0$ a.e and $|f_n-f|=(f_n – f)_+ + (fn-f)_-$ . If I can show $(f_n-f)_+ ≤ f$ and $(f_n-f)_-≤f$ . Then since $f_n$ and $f$ are integrable , by DCT, $\int|f_n-f| \to \int0 =0$ . I think my approach might be wrong ..
Best Answer
Hint:
Going to positive and negative parts is not really necessary. Here's another type of approach that is also quite straight-forward.
Denote $g_{n}=|f|+|f_{n}|-|f-f_{n}|$ for all $n$, which are non-negative (by triangle-inequality) and measurable. Apply Fatou's Lemma and use the fact that $\liminf(-a_{n})=-\limsup(a_{n})$.
If I calculated them correctly, then what you should get is $$\liminf_{n\to\infty}\int g_{n}=\|f\|-\limsup_{n\to\infty} \|f-f_{n}\|$$ and $$\int \liminf_{n\to\infty} g_{n} =\|f\|,$$
where $\|\cdot\|$ is the $L^{1}$-norm. Do you see how this implies your result?