  The aim of the manuscript is to characterize monotone metric in the space of
Markov map. Here, metric may not be Riemanian, or equivalently, may not be
induced from an inner product. So far, there have been plenty of literatures on
the metric in the space of probability distributions and quantum states. Among
them, Cencov and Petz characterized all the monotone metrics in the classical
and quantum state space. As for channels, however, only a little is known about
its geometrical structures. In that author's previous manuscript, the upper and
the lower bound of monotone channel metric was derived using resource
conversion theory, and it is proved that any monotone metric cannot be
Riemanian. . Due to the latter result, we cannot rely on Cencov's theory, to
build a geometric theory consistent across probability distributions and
channels. To dispense with the assumption that a metric is Riemanian, we
introduce some assumptions on asymptotic behavior;weak asymptotic additivity
and lower asymptotic continuity. The proof utilizes resource conversion
technique. In the end of the paper, an implication on quantum state metrics is
discussed.
