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By this Carathéodory's extension theorem, we know how to (uniquely) extend a pre-measure from a ring to a $\sigma$-field $\Sigma\subseteq2^{\Omega}$. So it is always posible to extend it further? From any $\sigma$-field, $\Sigma\subseteq\Gamma\subseteq2^\Omega$?

Also, the proof of "Lebesgue measure is invariant under translation" only proves for Borel-sets. And we know that there are sets that are Lebesgue measurable but not Borel. So is there any counter-examples?

  • to your second question, you can approximate lebesgue sets with borel sets to prove lebesgue measure is translation invariant for all lebesgue sets. – Mr. Cooperman Jan 10 '18 at 19:03
  • to your first question, you can always extend to the "completion" of a measure. And beyond that, https://math.stackexchange.com/questions/209532/extension-of-the-lebesgue-measurable-sets – Mr. Cooperman Jan 10 '18 at 19:05

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