I'm reading the following statement and I am confused about what the $\sup$ is over (it's not written in the book):
Let $H$ be the Hardy-Littlewood maximal function. A useful variant is: $$H^*f(x) = \sup\left\{ \frac{1}{m(B)}\int_B|f(y)|\,dy:B \text{ is a ball and } x \in B\right\}$$
What is the $\sup$ over? Isn't the term inside a fixed quantity? It's just the average on the ball of fixed size.