(Cross posted at [MSE](https://math.stackexchange.com/questions/3999840/is-there-a-maximal-translation-invariant-extension-of-lebesgue-measure).)


The answer to [this question](https://math.stackexchange.com/questions/209532/extension-of-the-lebesgue-measurable-sets) shows that there are translation-invariant extensions of Lebesgue measure.



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> Are there *maximal* translation-invariant extensions of Lebesgue measure (in ZFC)? If so, is there more than one?
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To be more precise, let $\mathcal L$ be the sigma-algebra of Lebesgue measurable subsets of $[0,1]$. By the linked post, there exists $A \subset [0,1]$ such that $A \notin \mathcal L$ and there exists a translation-invariant measure $\mu'$ on $\sigma(\mathcal L \cup \{A\})$ that extends the Lebesgue measure on $\mathcal L$. My question is:



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> Is there a sigma-algebra $\mathcal L' \supset \mathcal L$ and a measure $\mu'$ on $\mathcal L'$ such that $\mu'$ is a translation-invariant extension of Lebesgue measure and if $A \not \in \mathcal L'$, then there is no translation-invariant extension of $\mu'$ to $\sigma(\mathcal L' \cup \{A\})$? If such an $\mathcal L'$ exists, is it unique?
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Is there a maximal translation-invariant extension of Lebesgue measure?
The answer is no. This is the main result of "Extensions of invariant measures on Euclidean spaces" by Ciesielski and Pelc.


