Entropy, ultralimits and the Poisson boundary

In this paper, we introduce for a group G the notion of ultralimit of measure class preserving actions of it, and show that its Furstenberg-Poisson boundaries can be obtained as an ultralimit of actions on itself, when equipped with appropriately chosen measures. We use this result in embarking on a systematic quantitative study of the basic question of how close to invariant one can find measures on a G-space, particularly for the action of the group on itself. As applications, we show that on amenable groups there are always “almost invariant measures” with respect to the information-theoretic Kullback-Leibler divergence (and more generally, any f -divergence), making use of the existence of measures with trivial boundary. More interestingly, for a free group F and a symmetric measure λ supported on its generators, one can compute explicitly the infimum over all measures η on F of the Furstenberg entropy h_λ(F, η). Somewhat surprisingly, while in the case of the uniform measure on the generators the value is the same as the Furstenberg entropy of the Furstenberg-Poisson boundary of the same measure λ, in general it is the Furstenberg entropy of the Furstenberg-Poisson boundary of a measure on F different from λ.