A descending sequence with a non-additive measure where the measure of the limit isn't equal to the measure of the intersection

Question

Let $\mu^*$ be an outer measure over $\mathbb{R}$ so that $$\mu^*(A) := \begin{cases} 0 & \text{$A$ is finite or countable} \\ 1 & \text{otherwise} \end{cases}$$ A sequence of sets $(A_n)$ with $A_n \supseteq A_{n+1}$ and $A_i \subset \mathbb{R}$ should fulfill that $\lim_{n \to \infty} \mu^*(A_n)$ exists but $$\lim_{n \to \infty} \mu^*(A_n) \neq \mu^*\left(\bigcap_{k \in \mathbb{N}} A_n\right)$$ I know that $\mu^*$ is not additive so the equality isn't necesarily given by monotonous properties, but I'm missing a specific sequence that fulfills this.

Answer 1

Take $A_n = (-n^{-1}, n^{-1})$. Your LHS will always be $1$, whereas the RHS will be zero, as the $A_n$'s intersect at a single point.