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The definition of a product measure is given as

$$ (\mu_1\otimes\mu_2)(A) =\int_{\Omega_1} \mu_2(A_{\omega_1})\mu_1(d\omega_1) $$

This is the same as

$$ (\mu_1\otimes\mu_2)(A) =\int_{\Omega_1} \mu_2(A_{\omega_1})d\mu_1 $$

where $\omega_1 \in \Omega_1 \in \mathcal{A_1}$ with measure $\mu_1$.

Is this correct or does $\mu_1(d\omega_1) $ mean something else?

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