Geodesic
On the independence of subalgebras and decompositions
Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 30-38

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Conditions are investigated ensuring that a given system of subalgebras turns out to be independent after the given probability measure is replaced by an absolutely continuous one. 
@article{ZNSL_1979_85_a2,
     author = {D. A. Vladimirov},
     title = {On the independence of subalgebras and decompositions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {30--38},
     publisher = {mathdoc},
     volume = {85},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a2/}
}
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D. A. Vladimirov. On the independence of subalgebras and decompositions. Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 30-38. http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a2/