TITLE: Variation of a signed measure -
QUESTION [3 upvotes]: 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?

REPLY [1 votes]: 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.
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