Independent complements of subalgebras and product-measures
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 158-159

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For a complete Boolean algebra $\mathcal X$ conditions are given on the structure of subalgebras $\mathcal X^1$ and $\mathcal X^2$ which guarantee the existence on $\mathcal X$ of a strictly positive measure $\nu=\mu^1\times\mu^2$ such that the marginal measures $\mu^i$ are the projections of $\mu$ to $\mathcal X^i$.
@article{ZNSL_1986_149_a14,
     author = {A. A. Samorodnitskii},
     title = {Independent complements of subalgebras and product-measures},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {158--159},
     publisher = {mathdoc},
     volume = {149},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a14/}
}
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A. A. Samorodnitskii. Independent complements of subalgebras and product-measures. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 158-159. http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a14/