Hi

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

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

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

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

Help appreciated