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Let $(A_i,\mathcal{B}_i,\mu_i)$ for $i=1,2,\ldots$ be a sequence of probability spaces. Let $\nu_i$ be another sequence of probability measures on the same underlying measurable spaces. Assume that $\mu_i\ll \nu_i$ for any $i$. Under what conditions can we deduce that $\prod_i \mu_i \ll \prod_i \nu_i$?

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This is described by Kakutani's theorem on product measures. There is a very detailed exposition in Chapter III, Section 9 of Shiryaev's Probability.

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