## Prove result

Hi

Let $(X,\mathcal{M})$ be a measurable space and $(\mu_k)_{k=1}^{\infty}$ a sequence of positive measures on $\mathcal{M}$ such that $\mu_1 \leq \mu_2 \leq \mu_3 \leq ...$. Prove that the set function

$\mu(A) = \displaystyle\lim_{k \to \infty} \mu_k(A)$ is a positive measure.

By simply choosing $A=\emptyset$ we get $0 \leq 0 \leq 0 \leq ...$.

Now there is the linearity principle for any disjoint denumerable collection of members of $\mathcal{M}$.

Help appreciated