Proposition

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(Hj)_#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.

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