Markov kernels, convolution semigroups, and projective families of probability measures

For a measurable space (E,E ), we denote by E+ the set of functions E → [0,∞] that are E → B[0,∞] measurable. It can be proved that if I : E+ → [0,∞] is a function such that (i) f = 0 implies that I(f) = 0, (ii) if f, g ∈ E+ and a, b ≥ 0 then I(af + bg) = aI(f) + bI(g), and (iii) if fn is a sequence in E+ that increases pointwise to an element f of E+ then I(fn) increases to I(f), then there a unique measure μ on E such that I(f) = μf for each f ∈ E+. Let (E,E ) and (F,F ) be a measurable space. A transition kernel is a function K : E ×F → [0,∞] such that (i) for each x ∈ E, the function Kx : F → [0,∞] defined by B 7→ K(x,B) is a measure on F , and (ii) for each B ∈ F , the map x 7→ K(x,B) is measurable E → B[0,∞]. If μ is a measure on E , define