# Lebesgue Outer Measure Limit

Let $$\mu^{*}$$ be the Lebesgue outer measure on $$\mathbb{R}$$. I found the following exercise in a textbook.

Exercise. For every $$A \subseteq \mathbb{R}$$, $$\lim_{k \to \infty} \mu^{*}(A \cap [-k,k]) = \mu^{*}(A)$$.

I can get the case $$\mu^{*}(A) < \infty$$, but I am stuck with the case $$\mu^{*}(A) = \infty$$. Please help. Not homework, just reading.

Since the exercise appears before the introduction of measures, I presume that the countable additivity and continuity of the Lebesgue measure should not be used. This is in contrast to the answers: https://math.stackexchange.com/a/1577811/95800 https://math.stackexchange.com/a/117252/95800

Here is my argument for the case $$\mu^{*}(A) < \infty$$. There exists a sequence of intervals $$I_1,I_2,\ldots$$ such that $$A \subseteq \bigcup_{i=1}^{\infty} I_i$$ and $$\sum_{i=1}^{\infty} \ell(I_i) < \infty$$. Let $$\epsilon > 0$$ be given. There exists $$N \in \mathbb{N}$$ such that $$\sum_{i>N} \ell(I_i) < \epsilon$$. Choose $$K \in \mathbb{N}$$ such that $$\bigcup_{i=1}^{N} I_i \subseteq [-K,K]$$. Then $$A \setminus [-K,K] \subseteq \bigcup_{i > N} I_i$$. Moreover, $$A \setminus [-k,k] \subseteq \bigcup_{i > N} I_i$$ for every $$k \geq K$$. Therefore $$\mu^{*}(A) - \mu^{*}(A \cap [-k,k]) \leq \mu^{*}(A\setminus [-k,k]) \leq \sum_{i>N} \ell(I_i) < \epsilon$$ for every $$k \geq K$$. This proves $$\mu^{*}(A) = \lim_{k \to \infty} \mu^{*}(A \cap [-k,k])$$.