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 $A_1\in\mathcal A_1, A_2\in\mathcal A_2$: $$(P_1\otimes P_2) (A_1\times A_2) = P_1(A_1)P_2(A_2)?$$


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