[Merged by Bors] - feat: Prove that the measure equivalence between EuclideanSpace ℝ ι and ι → ℝ is volume preserving #7037
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We prove some lemmas that will be useful in following PRs #6832 and #7037, mainly: ```lean theorem Basis.addHaar_eq {b : Basis ι ℝ E} {b' : Basis ι' ℝ E} : b.addHaar = b'.addHaar ↔ b.addHaar b'.parallelepiped = 1 theorem Basis.parallelepiped_eq_map (b : Basis ι ℝ E) : b.parallelepiped = (TopologicalSpace.PositiveCompacts.piIcc01 ι).map b.equivFun.symm b.equivFunL.symm.continuous theorem Basis.addHaar_map (b : Basis ι ℝ E) (f : E ≃L[ℝ] F) : map f b.addHaar = (b.map f.toLinearEquiv).addHaar ``` Co-authored-by: Eric Wieser <[email protected]>
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We prove some lemmas that will be useful in following PRs #6832 and #7037, mainly: ```lean theorem Basis.addHaar_eq {b : Basis ι ℝ E} {b' : Basis ι' ℝ E} : b.addHaar = b'.addHaar ↔ b.addHaar b'.parallelepiped = 1 theorem Basis.parallelepiped_eq_map (b : Basis ι ℝ E) : b.parallelepiped = (TopologicalSpace.PositiveCompacts.piIcc01 ι).map b.equivFun.symm b.equivFunL.symm.continuous theorem Basis.addHaar_map (b : Basis ι ℝ E) (f : E ≃L[ℝ] F) : map f b.addHaar = (b.map f.toLinearEquiv).addHaar ``` Co-authored-by: Eric Wieser <[email protected]>
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…nd ι → ℝ is volume preserving (#7037) We prove that the two `MeasureSpace` structures on $\mathbb{R}^\iota$, the one coming from its identification with `ι→ ℝ` and the one coming from `EuclideanSpace ℝ ι`, agree in the sense that the measure equivalence between the two corresponding volumes is measure preserving. Co-authored-by: Eric Wieser <[email protected]> Co-authored-by: Eric Wieser <[email protected]>
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We prove some lemmas that will be useful in following PRs #6832 and #7037, mainly: ```lean theorem Basis.addHaar_eq {b : Basis ι ℝ E} {b' : Basis ι' ℝ E} : b.addHaar = b'.addHaar ↔ b.addHaar b'.parallelepiped = 1 theorem Basis.parallelepiped_eq_map (b : Basis ι ℝ E) : b.parallelepiped = (TopologicalSpace.PositiveCompacts.piIcc01 ι).map b.equivFun.symm b.equivFunL.symm.continuous theorem Basis.addHaar_map (b : Basis ι ℝ E) (f : E ≃L[ℝ] F) : map f b.addHaar = (b.map f.toLinearEquiv).addHaar ``` Co-authored-by: Eric Wieser <[email protected]>
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…nd ι → ℝ is volume preserving (#7037) We prove that the two `MeasureSpace` structures on $\mathbb{R}^\iota$, the one coming from its identification with `ι→ ℝ` and the one coming from `EuclideanSpace ℝ ι`, agree in the sense that the measure equivalence between the two corresponding volumes is measure preserving. Co-authored-by: Eric Wieser <[email protected]> Co-authored-by: Eric Wieser <[email protected]>
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We prove that the two$\mathbb{R}^\iota$ , the one coming from its identification with
MeasureSpacestructures onι→ ℝand the one coming fromEuclideanSpace ℝ ι, agree in the sense that the measure equivalence between the two corresponding volumes is measure preserving.Co-authored-by: Eric Wieser [email protected]