Let $(X,M,\mu)$ be a measure space, and let $p$ be number such that $1\le p < \infty$. Suppose $f_n, n \ge 1$, is a sequence of $M-$measuarable function on $X$ such that $f_n \rightarrow f$ $\mu-$almost everywhere on $X$. Suppose also that $|f_n(x)|\le g(x)$ for all $n\ge1$ and $x\in X$ , where $g\in L^p$.
How to prove that $\left\lVert f-f_n \right\rVert_p \rightarrow 0$ as $ n\rightarrow \infty$.
Any help appreciated.
You can prove this like you prove LDCT: use the basic inequality $|a+b|^p \le 2^p(|a|^p + |b|^p)$ to obtain $$ 2^p |f_n|^p + 2^p |f|^p - |f_n - f|^p \ge 0. $$ Since $|f_n| \le g$ almost everywhere and $f_n \to f$ almost everywhere, you also have $|f| \le g$ almost everywhere. Thus $$2^{p+1} g^p - |f_n - f|^p \ge 0. $$ Now apply the Fatou lemma: $$ \int_X 2^{p+1} g^p \, d\mu = \int_X \lim_{n \to \infty} \left( 2^{p+1} g^p - |f_n - f|^p \right) \, d\mu \le \liminf_{n \to \infty} \int_X \left( 2^{p+1} g^p - |f_n - f|^p \right) \, d\mu.$$ Some basic properties of the liminf imply $$\liminf_{n \to \infty} \int_X \left( 2^{p+1} g^p - |f_n - f|^p \right) \, d\mu = \int_X 2^{p+1} g^p \, d\mu - \limsup_{n \to \infty} \int_X |f_n - f|^p \, d\mu.$$ Thus $$\limsup_{n \to \infty} \int_X |f_n - f|^p \, d\mu \le 0.$$