I am studying measure theory and I am beginner and looking for some counter examples or guides to prove these questions: Let $\{\mu_n\}$ be a sequence of positive measures on $(X, M)$ which converges to $\mu$.
1-This sequence can be ascending or descending or an arbitrary sequence, in which case(s) $\mu$ is a measure?
2- If $\mu$ is a measure and all the $\mu_n$ are a complete measure, then $\mu$ is complete?
I think if this sequence be ascending or descending,$\mu$ is a measure, but I have no idea to start the proof. and also I do not have any intuition about completeness.
any help would be great thanks.
It seems that the convergence in question is $$\mu(E)=\lim_{n\to\infty}\mu_n(E)\quad(E\in M).$$
1.a. In general $\mu$ need not be a measure. For example, let $X=\Bbb R$, let $M$ be the algebra of Lebesgue measurable sets, and let $$\mu_n(E)=m(E\cap[n,\infty)),$$whhere $m$ is Lebesgue measure. If you figure out what this limit is for a few sets you can show that $$\mu(\Bbb R)\ne\sum_{k\in\Bbb Z}\mu([k,k+1)).$$
1.b. Note that $\mu_{n+1}\le\mu_n$ in that example. If $\mu_{n+1}\ge\mu_n$ then yes $\mu$ is a measure. Hint for that: If the $E_k$ are disjoint and $E=\bigcup E_k$ you can use Monotone Convergence to show that $$\mu(E)=\sum_k\mu(E_k).$$