There is a special non-commutative group related to the isometry ♯ : H / ♯ → P 2 ( R d ) , namely the set G ( Ω ) of Borel maps S : Ω → Ω (they lie in H ) that are almost everywhere invertible and have the same law as the identity map id.Here1/ Ω is the ball of unit volume in R d , centered at the origin.2. H := L 2 ( Ω , d x , R d ) My naïve guess is that "almost everywhere invertible" means the Lebesgue measure of { ω ∈ Ω ∣ card ⁡ ( S − 1 ( ω ) ) ≤ 1 } is 1.Could you elaborate on this notion?

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There is a special non-commutative group related to the isometry   ♯  :      H        /    ♯  →            P              2            (                  R                    d              )  , namely the set       G    (  Ω  ) of Borel maps   S  :  Ω  →  Ω (they lie in       H   ) that are almost everywhere invertible and have the same law as the identity map id.Here1/   Ω is the ball of unit volume in             R              d      , centered at the origin.2.       H    :=      L          2            (    Ω    ,          d        x    ,                  R                    d              )  My naïve guess is that &quot;almost everywhere invertible&quot; means the Lebesgue measure of   {  ω  ∈  Ω  ∣  card  ⁡  (      S          −      1        (  ω  )  )  ≤  1  } is 1.Could you elaborate on this notion?

Nirdaciw3 · Accepted Answer

When something happens almost everywhere, this means by definition that the set of points where it doesnt happens has zero measure. In your situation, the set of points where maps   S are not invertible has zero measure.  m  (  {  x  ∈  Ω      |    S  (  x  ) not invertible})=0.

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