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How to show uncountable Lebesgue measure
Let $(X,\mathcal{A},\mu)$ be a measure space. If $A$ is a measurable set, $\mu(A)$ denotes the Lebesgue measure of $A$. Define $\tilde{\mu}(A)=\mu(A)+\mu(\mathbb{R}\setminus A)$. Prove that for any measurable set $A\subseteq\mathbb{R}$, $\tilde{\mu}(A)=\infty$ if $A$ has a nonempty interior.
A:
Suppose $A$ has nonempty interior. Let $C=\operatorname{int}(A)$. Since $C$ is open, $C\setminus\{x_0\}$ is a nonempty open set for some $x_0\in C$. But $C$ has zero Lebesgue measure, so $\mu(C)=0$. And $C\subseteq A$, so $\mu(A)=\mu(C)=0$. Thus, $\tilde{\mu}(A)=1+0=1$.
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