# Show $\pi$ is a measure

Show that $\pi(E)=sup\lbrace \mu(A): A\subseteq E, A\in\mathbf{X} \rbrace$ is a measure on $\mathbf{X}$.

$\mu$ is a charge on $\mathbf{X}$ ($\sigma$-algebra), let $\pi$ be defined for $E\in\mathbf{X}$.

I have difficulty showing the $\sigma$-algebra countable additive property.

Found on wikipedia: A charge is a finitely additive, signed measure. What is X, an algebra? a sigma-algebra? It seems to me, if $\mu$ is a positive charge, then $\pi = \mu$ so this will not gain countable additivity for us. – GEdgar Mar 2 '14 at 13:43