Rational conditional distribution functions produce real regular kernels

Use the everywhere good filled data of [F1], writing them again $G_q$. For fixed $\omega$, put $F_{\omega }(x)={\inf }_{q\in \mathbb{Q},q>x}{G}_q(\omega )$. The index set is nonempty by F7 and countable by F8. Its values lie in $[0,1]$, and shrinking it as $x$ increases shows monotonicity. For rational $r$, monotonicity of the data gives $G_r\le F_{\omega }(r)\le G_{r+1\mathrm{/}n}$ for every positive integer $n$, so the rational right-limit condition gives $F_{\omega }(r)=G_r$. In particular $F_{\omega }(n)\to 1$ and $F_{\omega }(-n)\to 0$, and monotonicity gives both real tail limits.

If $x_j\downarrow x$, monotonicity gives a limit $L\ge F_{\omega }(x)$. For every rational $q>x$, eventually $x_j<q$, so $F_{\omega }(x_j)\le G_q(\omega )$ and $L\le G_q(\omega )$. Taking the infimum gives $L\le F_{\omega }(x)$. In particular the sequence $x+1\mathrm{/}j$ proves right continuity. By the existence clause of [F2] there is a Borel probability with CDF $F_{\omega }$. If $\mu ,\nu$ are two such probabilities, the equality class $\mathcal{C}=\{A:\mu (A)=\nu (A)\}$ contains $\mathbb{R}$ and every closed half-line, is closed under complements by subtraction from the common finite total one, and is closed under countable disjoint unions by countable additivity. It is a lambda-system containing the closed-half-line pi-system, whose generated sigma-algebra is Borel by [F4]; [F3] gives $\mu =\nu$. Thus the probability is locally proved unique and defines $K(\omega ,\cdot )$ without a family choice. On $N$ its CDF is ${\mathbf{1}}_{\{x\ge 0\}}$, so the same uniqueness identifies it with ${\delta }_0$.

For fixed real $x$, [F5] makes $F_{\omega }(x)$ a $\mathcal{G}$-measurable countable infimum. Using F7 and the fixed enumeration in F8, choose recursively the first enumerated rational $q_1\in (x,x+1)$ and then the first $q_{j+1}\in (x,\min (q_j,x+1\mathrm{/}(j+1)))$; thus $q_j\downarrow x$. Step 2.1 and rational agreement give $G_{q_j}\downarrow F_{\omega }(x)$. The bounded decreasing-convergence clause locally proved in [F1], applied on $H$, gives ${\int }_H{F}_{\omega }(x)dP={\lim }_j{\int }_H{G}_{q_j}dP={\lim }_{j}P(H\cap \{X\le q_j\})=P(H\cap \{X\le x\})\mathrm{.}$ The last indicator limit includes the point $X=x$.

Let $\mathcal{D}$ be the Borel sets $A$ for which $\omega \mapsto K(\omega ,A)$ is $\mathcal{G}$-measurable and the stated identity holds for every $H\in \mathcal{G}$. It contains $\mathbb{R}$. Complements use $1-K(\omega ,A)$ and subtraction from the finite total $P(H)$. For pairwise disjoint $A_j\in \mathcal{D}$, countable additivity of each section gives the union evaluation as the increasing limit of finite partial sums. This is measurable by [F5], and the locally proved monotone convergence in [F1] gives the event identity. Hence $\mathcal{D}$ is a lambda-system containing the closed-half-line pi-system from step 3.1. By [F3] and [F4] it contains every Borel set. The measurable evaluations and probability sections prove both kernel axioms and the full conditional identity.