A Yosida–Hewitt decomposition for totally monotone set functions on locally compact σ-compact topological spaces

We prove for totally monotone set functions defined on the set of Borel sets of a locally compact σ-compact topological space a similar decomposition theorem to the famous Yosida–Hewitt’s one for finitely additive measures. This way any totally monotone decomposes into a continuous part and a pathological part which vanishes on the compact subsets. We obtain as corollaries some decompositions for finitely additive set functions and for minitive set functions.