Suppose m is a measure on the Borel subsets B of a metric space S. Suppose for each t > 0 there exist jointly measurable non-negative functions p t : S × S → R such that   for each x and t and define...

Suppose m is a measure on the Borel subsets B of a metric space S. Suppose for each t > 0 there exist jointly measurable non-negative functions p_t: S × S → R such that

for each x and t and define

Show that the kernels P_t
satisfy the Chapman–Kolmogorov equations if and only if

for every s,t ≥ 0, every x ∈ S, and m-almost every z.