Simultaneous ae uniqueness of regular conditional distributions

Fix the countable determining algebra $\mathcal{A}$ from [F2], whose AC hypothesis is [F4]. For $A\in \mathcal{A}$, put $u=K(\cdot ,A)$ and $v=L(\cdot ,A)$. By [F1], both are $[0,1]$-valued and have equal integrals on every $H\in \mathcal{G}$. For $m\ge 1$, the set $H_m^{+}=\{u\ge v+1\mathrm{/}m\}$ belongs to $\mathcal{G}$. Monotonicity, additivity and positive homogeneity from [F3] give ${\int }_{H_m^{+}}udP\ge {\int }_{H_m^{+}}vdP+P(H_m^{+})\mathrm{/}m\mathrm{.}$ Equality of the two event integrals forces $P(H_m^{+})=0$. The same argument with $u,v$ interchanged shows that $\{v\ge u+1\mathrm{/}m\}$ is null. Their countable union is the measurable discrepancy set $N_A=\{u\mathrm{\ne }v\}$, so $N_A$ is null. This proves the needed scalar uniqueness locally, without importing conditional-expectation uniqueness.

The set $N={\bigcup }_{A\in \mathcal{A}}{N}_A$ is in $\mathcal{G}$ and null by countability and [F5]. For any $\omega \mathrm{\notin }N$ the two probability measures agree on every member of $\mathcal{A}$. Their total masses are finite, so the locally supplied determination assertion [F2] yields equality on the entire target sigma-algebra. This gives a single exceptional set independent of the target event.