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 $\prod\_i \mu\_i \ll \prod\_i \nu\_i$?



Absolute continuity of infinite product of probability measures
This is described by [Kakutani's theorem on product measures](https://mathscinet.ams.org/mathscinet-getitem?mr=23331). There is a very detailed exposition in Chapter III, Section 9 of Shiryaev's [Probability](https://mathscinet.ams.org/mathscinet-getitem?mr=1368405). 


