Variation of a signed measure -

I am studying Measure Theory, using the Bartle's book "Elements of Integration and Lebesgue Measure" and I couldn't do the exercise 3.Q:

"If $\mu$ is a charge on $X$, let $\mathcal{v}$ be defined for $E \in X$ by $$\mathcal{v}(E) = \sup \sum_{j=1}^n | \mu(A_j)|$$

Where the supremum is taken over all finite disjoint collections $\{ A_j\}$ in $X$ with $E=\cup_{j=1}^n A_j$. Show that $\mathcal{v}$ is a measure in $X$."

Ok, how can I prove that $\mathcal{v}$ is countably additive?

• What does charge mean? A countably additive signed measure? Commented Apr 17, 2013 at 18:09
• Yes, exactly. Sorry, my first book in this area, I have thought that was a common designation. Commented Apr 18, 2013 at 0:30

Hint: the collection $A_j$ need not be finite; you will get an equivalent definition if you use countable collections instead. Pick a countable collection of disjoint $E_j$ and try to show two inequalities $$\nu\left(\bigcup E_j\right)\leq/\geq\sum \nu(E_j)$$ instead of trying to show equality directly.