Let $(X,\mathcal{A},\mu)$ be a probability space.
Let $v \in \mathcal{M}^+(\mathcal{A})$.
Show that $|\int_X v \, d\mu | \leq ||v||$.
$||v||$ denotes the uniform norm.
I know the following but would like some help to connect it with the uniform norm given by $ ||v|| = \sup \lbrace |v(x)| : x \in X \rbrace $
$|\int_X v \, d\mu | \leq \int_X |v| \, d\mu $.
With $\|v\| =\sup \lbrace |v(x)| : x \in X \rbrace,$ calculate
$|\int_X v \, d\mu | \leq \int_X|v|d\mu\le \int_X\|v\| d\mu=$
$\|v\|\int_Xd\mu=\|v\|\cdot \mu (X)=\|v\|\cdot 1=\|v\|.$