in many important actions of groups there are no invariant measures. for example: the action of a free group on its boundary and the action of any discrete infinite group on itself. the problem we will discuss in this talk is 'on a given action, how invariant measure can be? '. our measuring of non-invariance will be based on entropy (f-divergence).

in the talk i will describe the solution of this problem for the free group acting on its boundary and on itself. for doing so we will introduce the notion of ultra-limit of $g$-spaces, and give a new description of the poisson-furstenberg boundary of $(g,k)$ as an ultra-limit of $g$ action on itself, with 'abel sum' measures. another application will be that amenable groups possess kl-almost-invariant measures (kl stands for the kullback-leibler divergence). all relevant notions, including the notion of poisson-furstenberg boundary and the notion of ultra-filters will be explained during the talk. this is a master thesis work under the supervision of yehuda shalom.