观点

Lemma 1. Let d≥2 be an integer, let B⊂^d be open, let ν be a Borel probability measure with ν(B)=1 , and let H_1,…,H_4:B→^d be homeomorphisms onto open images. Suppose that B,H_1(B),…,H_4(B) are pairwise disjoint. Define μ=13ν+16∑_ j=1 …

Lemma 1. Let d≥2 be an integer, let B⊂^d be open, let ν be a Borel probability measure with ν(B)=1 , and let H₁,…,H₄:B→^d be homeomorphisms onto open images. Suppose that B,H₁(B),…,H₄(B) are pairwise disjoint. Define μ=13ν+16∑_j=1⁴(H_j)_#ν. There is no Borel map T:^d→^d such that T_#μ=μ and T(x)∈H₁(x),…,H₄(x) ν -almost every x∈B.

证明 1证否 0作为观点被引用 2链条图发起推演

发起推演1

证明2026-10-21

Theorem 1.1. There is a nonnegative density ρ on ^3 such that ∫_ ^3 ρ(x) x=1,…观点

For each j , define the Borel set S_j= x∈B:T(x)=H_j(x) . Its image H_j(S_j) is Borel. Pairwise disjointness of the component images and injectivity of H_j give μ(S_j)=13ν(S_j), μ(H_j(S_j))=16ν(S_j). Since S_j⊂T^ -1 (H_j(S_j)) , preservation of μ would imply 13ν(S_j) ≤μ(T^ -1 (H_j(S_j))) =μ(H_j(S_j)) =16ν(S_j). Thus ν(S_j)=0 for all four indices, contradicting .

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