Measure kernel and probability kernel

A measure kernel from $(S,\mathrm{\Sigma })$ to $(T,\mathrm{T})$ is a map $K:S\times \mathrm{T}\to [0,\mathrm{\infty }]$ such that, for every $s\in S$, $A\mapsto K(s,A)$ is a measure on $(T,\mathrm{T})$, and for every $A\in \mathrm{T}$, $s\mapsto K(s,A)$ is $\mathrm{\Sigma }$-measurable. Measures and measurability have the meanings in Measures on sigma-algebras and A measurable function between measurable spaces; the evaluation functions use the Borel sigma-algebra on $[0,\mathrm{\infty }]$.

A probability kernel satisfies $K(s,T)=1$ for every s. A finite kernel satisfies $K(s,T)<\mathrm{\infty }$ for every s; there need not be a common bound on these masses. A uniformly sigma-finite kernel, in the convention of this page, comes with a specified sequence $T_n\in \mathrm{T}$ increasing to T such that $K(s,T_n)<\mathrm{\infty }$ for every s and n. This is a common measurable exhaustion, not a bound uniform in s. A finite kernel has the constant exhaustion $T_n=T$. An arbitrary measure kernel is not assumed to have such an exhaustion.

All these requirements are pointwise in s, not merely almost everywhere for an unspecified measure on S. If T is empty, its only measure is zero, so a probability kernel into T can exist only when S is empty. If S is empty, the kernel requirements are vacuous. The zero kernel is finite. This definition selects no versions and uses no choice axiom.