Let \((X, \mathscr{A}, \mu)\) be a finite measure space, \(1 \leqslant p<\infty\) and \(f \in \mathcal{L}^{p}\). Show that for \(m:=\int f d \mu\) and all \(a \in \mathbb{R}\) the following inequality holds:

\[2^{-p} \int|f(x)-m|^{p} \mu(d x) \leqslant \int|f(x)-a|^{p} \mu(d x)\]