Existence and Spectrality of random measures generated by infinite convolutions

In this paper, we construct a class of random measures μⁿ by infinite convolutions. Given infinitely many admissible pairs \(Nₖ, Bₖ)\ₖ₌₁∞ and a positive integral sequence n=ₖ\ₖ₌₁∞, for every ω∈ NN, we write μⁿ(ω) = δ_N_ω₁^-n₁B_ω₁ * δ_N_ω₁^-n₁N_ω₂^-n₂B_ω₂ * ⋯. If nₖ=1 for k≥ 1, write μ(ω)=μⁿ(ω). First, we show that the mapping μⁿ: (ω, B) ↦ μⁿ(ω)(B) is a random measure if the family of Borel probability measures \μ(ω) : ω ∈ NN\ is tight. Then, for every Bernoulli measure P on NN, the random measure μⁿ is also a spectral measure P-a.e.. If the positive integral sequence n is unbounded, the random measure μⁿ is a spectral measure regardless of the measures on the sequence space NN. Moreover, we provide some sufficient conditions for the existence of the random measure μⁿ. Finally, we verify that random measures have the intermediate-value property.