Strong Lp bound for the Hardy-Littlewood maximal function ($\Vert Mf{\Vert }_p\le C_p\Vert f{\Vert }_p$)

For $1<p\le \infty$, the Hardy-Littlewood maximal function is bounded on $L^p({\mathbb{R}}^n)$. A direct proof for finite $p$ applies the weak $(1,1)$ estimate to $f{1}_{\{|f|>s/2\}}$, integrates the resulting distribution-function bound against $p{s}^{p-1}$, and uses Tonelli theorem.