I am following the book Probability Theory by A.Klenke. The exercise 1.3.3 is as follows:
Let $(\mu_n)_{n\in\mathbb{N}}$ be finite measures on a measurable space $(\Omega,\mathcal{A})$ such that for all $n\in\mathbb{N}$, for all $A\in\mathcal{A}$, there exists $\mu(A)=\lim\limits_{n\to\infty}\mu_n(A)$.
Show that $\mu$ is a measure. Hint: you can show that $\mu$ is $\emptyset$-continuous.
I have showed that $\mu$ is a content and is finite by definition. Also, $\mu(\emptyset)=0$. But I am struggling to show that $\mu$ is $\sigma$-additive by showing that it is $\emptyset$-continuous (as $\mu$ is finite).
My attempt:
Let $A_i\downarrow\emptyset$ with $A_i\supset A_{i+1}$ and $\bigcap_{i}A_i=\emptyset$ with $\mu(A_i)\neq\infty$ for all $i$.
We can write
$$
\mu(\emptyset)=\lim\limits_{n\to\infty}\lim_{i\to\infty}\mu_n(A_i)
$$
but we need a thing like uniform boundedness to swap the limits and the book has not tackled that yet (and still, this would require a bit of work I suppose as I have no idea where to begin).
Any ideas ?
Thank you !
Best Answer
As Kavi mentionned, the proof of the theorem is on the Wikipedia page.
A more elementary proof can actually be found here on Maths SE, I did not find it at first. I mark this post as resolved.