 Uncentered maximal function of a finite measure

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.