Absolute continuity of information channels |
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Authors: | Hisaharu Umegaki |
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Institution: | Tokyo Institute of Technology, Meguro-ku, Ohokayama, Tokyo, Japan |
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Abstract: | 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 ν(·,·) 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 μ on (Y,
) which dominates every measure ν(x, ·) (x X). Furthermore, under this condition, the Radon-Nikodym derivative ν(x, dy)/μ(dy) is jointly measurable with respect to the product measure space (X,
, m) (Y,
, μ) 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. |
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Keywords: | Channels Information sources absolute continuities strong measurabilities Radon-Nikodym derivative Hilbert-Schmidt Channels |
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