# Is there a maximal translation-invariant extension of Lebesgue measure?

(Cross posted at MSE.)

The answer to this question shows that there are translation-invariant extensions of Lebesgue measure.

Are there maximal translation-invariant extensions of Lebesgue measure (in ZFC)? If so, is there more than one?

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:

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?