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We study here the topology of information on the space of probabilitymeasures over Polish spaces that was defined in Hellwig (1996). We show thatunder this topology, a convergent sequence of probability measures satisfying aconditional independence property converges to a measure that also satisfiesthe same conditional independence property. This also corrects the proof of aclaim in Hellwig (1996, Lemma 4). Additionally, we determine sufficientconditions on the Polish spaces and the topology over measures spaces underwhich a convergent sequence of probability measures is also convergent in thetopology of information.

The Topology of Information on the Space of Probability Measures over Product of Polish Spaces