Geodesic
A Class of Singular Functions
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1425-1431

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Kakutani (2) has proved a very general theorem, giving necessary and sufficient conditions for two infinite product measures to be mutually absolutely continuous. To formulate Kakutani's result, let us first recall that a measurable space is a pair (E, B), where B denotes a Borel field (also called σ-ring) of subsets of E, and a measure m on this space is a countably additive set function on B (see Halmos (1)). 
Shapiro, Harold S. A Class of Singular Functions. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1425-1431. doi: 10.4153/CJM-1968-143-9
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[1] 1. Halmos, P., Measure theory (Van Nostrand, Princeton, N.J., 1950).10.1007/978-1-4684-9440-2 Google Scholar | DOI

[2] 2. Kakutani, S., Oequivalence of infinite product measures, Ann. of Math. (2) 1+9 (1948), 214–226 Google Scholar

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