There is a special non-commutative group related to the isometry $\mathrm{\u266f}:\mathbb{H}/\mathrm{\u266f}\to {\mathcal{P}}_{2}\left({\mathbb{R}}^{d}\right)$, namely the set $\mathcal{G}(\mathrm{\Omega})$ of Borel maps $S:\mathrm{\Omega}\to \mathrm{\Omega}$ (they lie in $\mathbb{H}$ ) that are almost everywhere invertible and have the same law as the identity map id.

Here

1/ $\mathrm{\Omega}$ is the ball of unit volume in ${\mathbb{R}}^{d}$, centered at the origin.

2. $\mathbb{H}:={L}^{2}(\mathrm{\Omega},\mathrm{d}x,{\mathbb{R}}^{d})$

My naïve guess is that "almost everywhere invertible" means the Lebesgue measure of $\{\omega \in \mathrm{\Omega}\mid \mathrm{card}({S}^{-1}(\omega ))\le 1\}$ is 1.

Could you elaborate on this notion?

Here

1/ $\mathrm{\Omega}$ is the ball of unit volume in ${\mathbb{R}}^{d}$, centered at the origin.

2. $\mathbb{H}:={L}^{2}(\mathrm{\Omega},\mathrm{d}x,{\mathbb{R}}^{d})$

My naïve guess is that "almost everywhere invertible" means the Lebesgue measure of $\{\omega \in \mathrm{\Omega}\mid \mathrm{card}({S}^{-1}(\omega ))\le 1\}$ is 1.

Could you elaborate on this notion?