  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 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 $\lambda$ supported on its generators,
one can compute explicitly the infimum over all measures $\eta$ on $F$ of the
Furstenberg entropy $h_{\lambda}(F,\eta)$. 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
$\lambda$, in general it is the Furstenberg entropy of the Furstenberg-Poisson
boundary of a measure on $F$ different from $\lambda$.
