Mutual absolute continuity of probability measures on a filtered space holds if and only if a certain martingale limit M equals 1 almost surely, and then the square roots of the Radon-Nikodym derivatives converge in L^2.
Durrett, Probability: theory and examples
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A characterization of mutual absolute continuity of probability measures on a filtered space
Mutual absolute continuity of probability measures on a filtered space holds if and only if a certain martingale limit M equals 1 almost surely, and then the square roots of the Radon-Nikodym derivatives converge in L^2.