I've seen a similar question on this forum before, yet i could not find a solution to this particular problem. Could you give me some tips on how to proceed with the solution? Here's the statement of the problem: Let $A$ be a bounded set in $\mathbb R^k$. Let $K$ be a $G_\delta$ set containing $A$, such that the outer measure of $A$ is equal the measure of $K$. Prove that $A$ is measurable.
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For any set $A \subset \mathbb R^n$ you can find a $G_\delta$ subset containing $A$ of the same outer measure. This follows because you can always find an open set $U$ containing $A$ such that $m^\ast(U)<m^\ast(A)+\varepsilon$, this is by the definition of outer measure. So by taking appropriate intersections you can find such a $G_\delta$ set. In particular your question is false because there exist non-measurable subsets.