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My question is does inner regularity of outer measure holds true for $\mathbb{R}$. For every $\epsilon>0$ can we get compact set $K$ such that $ m(\mathbb{R}-K)<\epsilon$ , if yes then which kind of compact set would work in this case.

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    $\begingroup$ $m(\mathbb R \setminus K)=\infty$ for any compact set $K$. $\endgroup$ Commented May 27, 2021 at 6:18

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This approximation of a Borel set by a compact set is in general true if the measure is Radon (the Lebesgue measure is one), if the space is locally compact and separable (the real line is), AND if the Borel set we are approximating has finite measure. This is not the case for the real line with the Lebesgue measure.

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