The existence of subspace wavelet sets

Let H be a reducing subspace of L2(Rd), that is, a closed subspace of L2(Rd) with the property that f(Amt−ℓ)∈H for any f∈H, m∈Z and ℓ∈Zd, where A is a d×d expansive matrix. It is known that H is a reducing subspace if and only if there exists a measurable subset M of Rd such that AtM=M and F(H)=L2(Rd)·χM. Under some given conditions of M, it is known that there exist A-dilation subspace wavelet sets with respect to H. In this paper, we prove that this holds in general.