Uncentered maximal function of a finite measure
= Uncentered maximal function of a finite measure
{title2=$M_u\mu(x)=\sup_{x\in B}\mu(B)/|B|$}
The uncentered maximal function takes the supremum over all open Euclidean balls containing $x$, not only balls centred at $x$. It is an extended nonnegative <lower semicontinuous> function: each strict superlevel set is the union of the balls whose average density exceeds that level. For finite positive measures it satisfies the <uncentered maximal weak-type inequality>.