= Solution
The centered <Hardy-Littlewood maximal function> is
$$
Mf(x)=\sup_{r>0}\frac1{2r}\int_{x-r}^{x+r}|f(y)|dy.
$$
The <Hardy-Littlewood maximal inequality> gives, for example,
$$
\boxed{\lambda(\{Mf>t\})\leq\frac3t\|f\|_1\quad(t>0).}
$$
This valid constant follows from the disjoint-interval covering argument: a compact subset of the superlevel set has a finite cover by intervals of average above $t$; select disjoint intervals whose triples cover it, bound their total length by $\|f\|_1/t$, and use <inner regularity>. An optimal constant is not required.
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