# Given two commuting projection-valued measures, is there a product measure?

Let $$H$$ be a Hilbert space. Let $$(\Omega_1,\mathcal A_1)$$ and $$(\Omega_2,\mathcal A_2)$$ be measurable spaces. Let $$P_1: \mathcal A_1\to\mathfrak L(H), \quad P_2: \mathcal A_2\to\mathfrak L(H)$$ be projection-valued measures. Assume that $$[P_1(A_1),P_2(A_2)] = 0$$ for all $$A_1\in\mathcal A_1, A_2\in\mathcal A_2$$. Let $$\mathcal A_1\otimes\mathcal A_2 := \sigma(\{A_1\times A_2|A_1\in\mathcal A_1, A_2\in\mathcal A_2\})$$

Is there a projection-valued measure $$P_1\otimes P_2: \mathcal A_1\otimes\mathcal A_2\to\mathfrak L(H)$$ such that for all $$A_1\in\mathcal A_1, A_2\in\mathcal A_2$$: $$(P_1\otimes P_2) (A_1\times A_2) = P_1(A_1)P_2(A_2)?$$