can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence?

Question

Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce that $f_n\to f$ almost everywhere?

Answer 1

For any $x\in\Omega$, the monotonic sequence $(f_n(x))$ has a limit, possibly equal to $-\infty$ or $+\infty$; so the sequence $(\vert f_n(x)-f(x)\vert)$ has a limit in $[0,\infty]$. Let $g$ be the measurable function (with values in $[0,\infty]$) defined by $g(x)=\lim_{n\to\infty} \vert f_n(x)-f(x)\vert^p$. By Fatou's lemma and since $\Vert f_n-f\Vert_p\to 0$, we have $$\int_\Omega g\, d\mu\leq \liminf_{n\to\infty}\int_\Omega \vert f_n-f\vert^pd\mu =0\, .$$ Since $g\geq 0$, it follows that $g(x)=0$ almost everywhere; in other words, $f_n(x)\to f(x)$ a.e.