Composition of probability kernels

For probability kernels $K:(S,\mathrm{\Sigma })\to (T,\mathrm{T})$ and $L:(T,\mathrm{T})\to (U,\mathrm{{\rm Y}})$, their composition, in the order K then L, is $(KL)(s,A)={\int }_{T}L(t,A)K(s,dt),s\in S,A\in \mathrm{{\rm Y}}\mathrm{.}$

The kernel convention is Measure kernel and probability kernel. The integrand is measurable in t and lies in $[0,1]$, so the displayed number exists in $[0,1]$. As a function of s it is measurable by Measurability of integration against a kernel, applied to the product-measurable function $(s,t)\mapsto L(t,A)$: its threshold preimages are rectangles $S\times \{t:L(t,A)>a\}$. The probability-kernel assertion and associativity are justified by Kernel composition is well defined and associative ↗.

Composition refers to specified pointwise kernels. Almost-everywhere classes alone do not define this formula until representatives and the relevant measures are specified. For an empty source the candidate is the empty map. No choice of versions or assumption of AC enters this definition.