  We study in this paper real-valued functions on the space of all
sub-$\sigma$-algebras of a probability measure space, and introduce the notion
of Kudo-continuity, which is an a priori strengthening of continuity with
respect to strong convergence. We show that a large class of entropy
functionals are Kudo-continuous. On the way, we establish upper and lower
continuity of various entropy functions with respect to asymptotic second order
stochastic domination, which should be of independent interest. An application
to the study of entropy spectra of $\mu$-boundaries associated to random walks
on locally compact groups is given.
