I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following exercise is Exercise 10 on p.46 in Exercises 2C in this book.
Exercise 10
Give an example of a measure space $(X,\mathcal{S},\mu)$ and a decreasing sequence $E_1\supset E_2\supset\cdots$ of sets in $\mathcal{S}$ such that $$\mu(\bigcap_{k=1}^\infty E_k)\neq\lim_{k\to\infty}\mu(E_k).$$
My solution is here:
Let $X:=\mathbb{R}.$
Let $\mathcal{S}:=2^X.$
Let $\mu:\mathcal{S}\to [0,\infty]$ be a function such that $\mu(E)=n$ if $E$ is a finite set containing exactly $n$ elements and $\mu(E)=\infty$ if $E$ is not a finite set.
Then, $(X,\mathcal{S},\mu)$ is a measure space. (Please see Example 2.55 on p.41 in the book.)
Let $E_k:=(0,\frac{1}{k}).$
Then, $E_1\supset E_2\supset\cdots$ holds.
$\mu(\bigcap_{k=1}^\infty E_k)=\mu(\emptyset)=0.$
$\lim_{k\to\infty}\mu(E_k)=\lim_{k\to\infty}\infty=\infty.$
So, $\mu(\bigcap_{k=1}^\infty E_k)\neq\lim_{k\to\infty}\mu(E_k).$
Is my solution ok or not?