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