The pushforward to the coset space G ⧸ Γ of the restriction of a both left- and right-invariant measure on an additive topological group G to a fundamental domain 𝓕 is a G-invariant measure on G ⧸ Γ.

The pushforward to the coset space G ⧸ Γ of the restriction of a both left- and right- invariant measure on G to a fundamental domain 𝓕 is a G-invariant measure on G ⧸ Γ.

Assuming Γ is a normal subgroup of an additive topological group G, the pushforward to the quotient group G ⧸ Γ of the restriction of a both left- and right-invariant measure on G to a fundamental domain 𝓕 is a left-invariant measure on G ⧸ Γ.

Assuming Γ is a normal subgroup of a topological group G, the pushforward to the quotient group G ⧸ Γ of the restriction of a both left- and right-invariant measure on G to a fundamental domain 𝓕 is a left-invariant measure on G ⧸ Γ.

Given a normal subgroup Γ of an additive topological group G with Haar measure μ, which is also right-invariant, and a finite volume fundamental domain 𝓕, the pushforward to the quotient group G ⧸ Γ of the restriction of μ to 𝓕 is a multiple of Haar measure on G ⧸ Γ.

Given a normal subgroup Γ of a topological group G with Haar measure μ, which is also right-invariant, and a finite volume fundamental domain 𝓕, the pushforward to the quotient group G ⧸ Γ of the restriction of μ to 𝓕 is a multiple of Haar measure on G ⧸ Γ.

Given a normal subgroup Γ of an additive topological group G with Haar measure μ, which is also right-invariant, and a finite volume fundamental domain 𝓕, the quotient map to G ⧸ Γ is measure-preserving between appropriate multiples of Haar measure on G and G ⧸ Γ.

Given a normal subgroup Γ of a topological group G with Haar measure μ, which is also right-invariant, and a finite volume fundamental domain 𝓕, the quotient map to G ⧸ Γ is measure-preserving between appropriate multiples of Haar measure on G and G ⧸ Γ.

The ess_sup of a function g on the additive quotient space G ⧸ Γ with respect to the pushforward of the restriction, μ_𝓕, of a right-invariant measure μ to a fundamental domain 𝓕, is the same as the ess_sup of g's lift to the universal cover G with respect to μ.

The ess_sup of a function g on the quotient space G ⧸ Γ with respect to the pushforward of the restriction, μ_𝓕, of a right-invariant measure μ to a fundamental domain 𝓕, is the same as the ess_sup of g's lift to the universal cover G with respect to μ.

Given a quotient space G ⧸ Γ where Γ is countable, and the restriction, μ_𝓕, of a right-invariant measure μ on G to a fundamental domain 𝓕, a set in the quotient which has μ_𝓕-measure zero, also has measure zero under the folding of μ under the quotient. Note that, if Γ is infinite, then the folded map will take the value ∞ on any open set in the quotient!

Given an additive quotient space G ⧸ Γ where Γ is countable, and the restriction, μ_𝓕, of a right-invariant measure μ on G to a fundamental domain 𝓕, a set in the quotient which has μ_𝓕-measure zero, also has measure zero under the folding of μ under the quotient. Note that, if Γ is infinite, then the folded map will take the value ∞ on any open set in the quotient!

This is a simple version of the Unfolding Trick: Given a subgroup Γ of an additive group G, the integral of a function f on G with respect to a right-invariant measure μ is equal to the integral over the quotient G ⧸ Γ of the automorphization of f.

This is a simple version of the Unfolding Trick: Given a subgroup Γ of a group G, the integral of a function f on G with respect to a right-invariant measure μ is equal to the integral over the quotient G ⧸ Γ of the automorphization of f.

This is the Unfolding Trick: Given a subgroup Γ of a group G, the integral of a function f on G times the lift to G of a function g on the quotient G ⧸ Γ with respect to a right-invariant measure μ on G, is equal to the integral over the quotient of the automorphization of f times g.

This is the Unfolding Trick: Given an additive subgroup Γ' of an additive group G', the integral of a function f on G' times the lift to G' of a function g on the quotient G' ⧸ Γ' with respect to a right-invariant measure μ on G', is equal to the integral over the quotient of the automorphization of f times g.