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We study in this paper real-valued functions on the space of allsub-$\sigma$-algebras of a probability measure space, and introduce the notionof Kudo-continuity, which is an a priori strengthening of continuity withrespect to strong convergence. We show that a large class of entropyfunctionals are Kudo-continuous. On the way, we establish upper and lowercontinuity of various entropy functions with respect to asymptotic second orderstochastic domination, which should be of independent interest. An applicationto the study of entropy spectra of $\mu$-boundaries associated to random walkson locally compact groups is given.

Kudo-Continuity Of Entropy Functionals