Example of set, finite outer measure, subsets, where outer measure does not converge

Question

What is an example of a set $X$ and a finite outer measure $\mu^*$ on $X$, subsets $A_n \uparrow A$ of $X$, and subsets $B_n \downarrow B$ of $X$ such that $\mu^*(A_n)$ does not converge to $\mu^*(A)$ and $\mu^*(B_n)$ does not converge to $\mu^*(B)$?

Answer 1

Let $X = \mathbb{R}$ and $\mu^{\star}(A) = 0$ if $A$ is countable, $\mu^{\star}(A) = 1$ if both $A$ and $\mathbb{R} \setminus A$ are uncountable and $\mu^{\star}(A) = 2$ if $\mathbb{R} \setminus A$ is countable. Check that $\mu^{\star}$ is an outer measure.

Put $A_n = [-n, n]$, $B_n = [n, \infty)$ and check.