Absolute continuity of information channels

Let [X, v, Y] be an abstract information channel with the input X = (X, ) and the output Y = (Y, ) which are measurable spaces, and denote by L(Y) = L(Y, ) the Banach space of all bounded signed measures with finite total variation as norm. The channel distribution [nu](·,·) is considered as a function defined on (X, ) and valued in L(Y). It will be proved that, if the measurable space (Y, ) is countably generated, then the is a strongly measurable function from X into L(Y) if and only if there exists a probability measure [mu] on (Y, ) which dominates every measure [nu](x, ·) (x [set membership, variant] X). Furthermore, under this condition, the Radon-Nikodym derivative [nu](x, dy)/[mu](dy) is jointly measurable with respect to the product measure space (X, , m) [circle times operator] (Y, , [mu]) where m is any but fixed probability measure of (X, ). As an application, it will be shown that the channel given as above is uniformly approximated by channels of Hibert-Schmidt type.