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Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall consider on it the following strong topology: a map $T_{j}$ converges to $T$ in $G[\mathbb T^2, \mathcal L^2]$ if $T_{j}$ and $T_{j}^{-1}$ converge uniformly on $\mathbb T^2$ to $T$ and $T^{-1}$.

How can we build a measure on this group that has the following property?

(1) A measure zero set has empty interior;

(2) Every subset of a measure zero set has measure zero too;

(3) Countable union of measure zero sets has measure zero too;

(4) quasi-invariance in some sense

Point (4) is deliberately a bit vague. The question is motivated by the related Building random homeomorphisms of the torus $\mathbb T^2$

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    $\begingroup$ Which topology are you putting on your group, and which sigma algebra are you choosing to define your measure on? $\endgroup$
    – Yemon Choi
    Commented Sep 6, 2022 at 1:35
  • $\begingroup$ @YemonChoi Added: we shall consider the following topology: a map $T_{j}$ converges to $T$ in $G[\mathbb T^2, \mathcal L^2]$ if $T_{j}$ and $T_{j}^{-1}$ converge uniformly on $\mathbb T^2$ to $T$ and $T^{-1}$. $\endgroup$
    – user490373
    Commented Sep 6, 2022 at 8:18
  • $\begingroup$ @YemonChoi For the measure to be constructed on the group, I don't know -- whichever one is more convenient for the desired properties to hold. $\endgroup$
    – user490373
    Commented Sep 6, 2022 at 8:24

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