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For a finite set D ⊂ Z and an integer b ≥ 2, we say thata∈E δ a denote the uniformly discrete probability measure on E. We prove that the class of infinite convolution (Moran measure) μ b,{Dk } = δ b −1 D1 * δ b −2 D2 * · · · is a spectral measure provided that there is a common C ⊂ Z + compatible to all the (b, D k ) and C + C ⊆ {0, 1, . . . , b − 1}. We also give some examples to illustrate the hypotheses and results, in particular, the last condition on C is essential.

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Let μ be a Borel probability measure with compact support in R 2 . μ is called a spectral measure if there exists a countable set Λ ⊂ R 2 such that E Λ = {e −2πi λ,x : λ ∈ Λ} is an orthonormal basis for L 2 (μ). In this note we prove that the integral Sierpinski measure μ A, D is a spectral measure if and only if (A, D) is admissible. This completely settles the spectrality of integral Sierpinski measures in R 2 .

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