Prove that $\lim_{n \rightarrow \infty} \int_{\Omega} |f_n - f|^{p} dm=0$.

Question

I am stuck on the following problem.

Let $1<p<\infty$ and let $\Omega \subset \mathbb{R}$. Let {$f_n$} be a sequence of measurable functions on $\Omega$ such that $\lim_{n \rightarrow \infty} f_n= f$ (a.e.) on $\Omega$. Suppose there exists a nonnegative measurable function $g$ on $\Omega$ such that $g^p \in L^{1} \Omega$ and $|f_n| \leq g$ (a.e.) on $\Omega$ for every $n \geq 1$. Prove that $\lim_{n \rightarrow \infty} \int_{\Omega} |f_n - f|^{p} dm=0$.

Plan:

Want to show that $|f_n-f|^p \leq g$ (a.e.), and then by the dominated convergence theorem, $\lim_{n \rightarrow \infty} \int_{\Omega} |f_n - f|^{p} dm=\int_{\Omega} \lim_{n \rightarrow \infty} |f_n-f|^p dm=0$.

Attempt:

Consider $|f_n-f| \leq |f_n|-|f| \leq g-|f| \leq g$. So $|f_n-f| \leq g$, raising both sides to the power $p$, $|f_n-f|^p \leq g^p \in L^{1} \Omega$ where $g^p$ is our control function thus allowing us to use dominated convergence theorem. Valid since $p >1$.

Hence, $\lim_{n \rightarrow \infty} \int_{\Omega} |f_n - f|^{p} dm=\int_{\Omega} \lim_{n \rightarrow \infty} |f_n-f|^p dm$. Since $lim_{n \rightarrow \infty} f_n=f \Leftrightarrow lim_{n \rightarrow \infty} |f_n-f| =0$ we have $\int_{\Omega} \lim_{n \rightarrow \infty} |f_n-f|^p dm=0$.

Any feedback on my attempt and hints would be very helpful. Thank you.

Answer 1

You're on the right track in that you want to apply DCT however, $\lvert x - y \rvert \not\le \lvert x \rvert - \lvert y \rvert$. Consider $x = 1, y = -1$, e.g.

Instead, since $f \to f_n$ a.e., we have $\lvert f - f_n \rvert^p \to 0$ a.e.; indeed, by definition $f_n \to f$ is the same as $\lvert f - f_n \rvert \to 0$ and then $\lvert f -f_n \rvert^p \to 0$ since $t \mapsto t^p$ is continuous. Next, $$\lvert f - f_n \rvert^p \le (\lvert f \rvert + \lvert f_n \rvert)^p \le (2\max\{\lvert f \rvert, \lvert f_n \rvert\})^p =2^p \max\{\lvert f \rvert^p, \lvert f_n \rvert^p\} \le 2^p(\lvert f \rvert^p + \lvert f_n \rvert^p). $$ Now, $\lvert f \rvert \le g$ a.e. since $\lvert f \rvert = \lim \lvert f_n \rvert \le \lim g = g$ a.e. Thus $$\lvert f - f_n \rvert^p \le 2^{p+1} g^p.$$ However, you have assumed that $g^p$ is integrable, so by DCT you have $$\lim \int_\Omega \lvert f - f_n \rvert dm = \int_\Omega \lim \lvert f - f_n \rvert dm = \int_\Omega 0 \, dm = 0.$$