Let $(X, \mathcal{A}, \mu)$ be a dimensional space, $\left(f_j\right)_{j \subset \mathbb{N}} \subsetneq \mathfrak{L}^1(X)$ a sequence of integrable functions and $f: X \rightarrow \mathbb{R}$ a function such that $f_j \stackrel{j \rightarrow \infty}{\longrightarrow} f$ $\mu$-almost everywhere. Let $C>0$ be given such that $$ \int\left|f_j(x)\right| d \mu(x)<C $$ for all $j \in \mathbb{N}$. Show that $$ \lim _{j \rightarrow \infty} \int|| f_j|-| f|-| f_j-f \|=0 $$ is valid. Hint: First prove that ||$f_j|-| f|-| f-f_j|| \leq 2|f|$ holds.
The Hint: ||$f_j|-| f|-| f-f_j|| \leq ||f_j-f|-|f-fj|| \leq |2f_j-2f| \leq 2|f|$
Now there is a majorant. So we can use Lebesgue's theorem of dominated convergence. Than we can put the lim inside. Then the integral is 0 because fj is valid after f. And although fj to f only works almost everywhere, because fj is bounded.
Is it correct?
(b) Conclude that $$ \int|f|=\int\left|f_j\right|-\int\left|f-f_j\right|+a_j $$ where $a_j \rightarrow 0$ converges. Compare this statement with Fatou's lemma.
How can i do this with a)? Can i use the Linearity?