I have a (possibly) very stupid question. I don't understand why the supremum is used instead of the maximum. Here's my problem.
We denote by $\mathcal{H}(M,\omega)$ the set of all smooth functions $H : M \rightarrow \mathbb{R}$ satisfying the following properties.
$\exists$ an open subset $U \subset M$ and a compact set $K \subset M$ such that $U \subset K \subset M \setminus \partial M$ and satisfying
$H_{\big|U} \equiv 0$ is constant.
$H_{\big|M \setminus K} \equiv m(H)$ is constant.
$0 \leq H(x) \leq m(H), \forall x \in M$.
The constant $m(H) := maxH - minH $ is called the oscillation of the function $H$.
Now define the following object, called a capacity.
$c_0 := sup \{ m(H) \mid H \in \mathcal{H}(M,\omega) \}$. Why do we need to use the $sup$ ? Could we not directly use the $max$ ?
Thanks for your help.