Brezis excercise 4.3.1. If $f$ and $g$ are in $L^p(\Omega)$, then $h = max\{f(x),g(x)\}$ is in $L_p(\Omega)$.

Question

Given two functions $f,g\in L_p(\Omega)$, define $h(x)=max\{f(x),g(x)\}$. Prove that $h\in L_p(\Omega)$.

I know that $|h(x)|^p\geq |f(x)|^p$ and $|h(x)|^p\geq |g(x)|^p$ but this seems to go nowhere... any hints?

Answer 1

Hint: for all $x\in\Omega$, the following inequalities hold: $$ \left\lvert h(x)\right\rvert^p\leqslant \max\left\{\left\lvert f(x)\right\rvert^p,\left\lvert g(x)\right\rvert^p\right\}\leqslant \left\lvert f(x)\right\rvert^p+\left\lvert g(x)\right\rvert^p. $$