Standard borel spaces have countable generating and measure determining algebras

Fix $f$ from [F1]. Enumerate the pullbacks $H_q=f^{-1}[B\cap (-\mathrm{\infty },q]]$ using [F2]. Let ${\mathcal{A}}_n$ be the Boolean algebra on the first $n$ pullbacks, with ${\mathcal{A}}_0=\{\mathrm{\varnothing },E\}$. Its atoms are the at most $2^n$ intersections obtained by taking each generator or its complement; every member is a union of atoms. Thus each ${\mathcal{A}}_n$ is finite, and $\mathcal{A}={\bigcup }_n{\mathcal{A}}_n$ is an algebra: any two elements lie in a common ${\mathcal{A}}_n$.

AC implies [F5], so [F4] makes $\mathcal{A}$ countable. This is the exact countable-choice use for enumerating the finite algebras. The trace of the generators of [F3] generates $\mathcal{B}(B)$, so bimeasurability of $f$ gives $\sigma (\mathcal{A})=\mathcal{S}$. If $x\mathrm{\ne }y$, injectivity gives different codes; [F8] provides a rational between them, and its pullback contains exactly the lower-coded point.

For finite $\mu ,\nu$ agreeing on $\mathcal{A}$, their total masses agree because $E\in \mathcal{A}$. The equality class $\mathcal{D}=\{A\in \mathcal{S}:\mu (A)=\nu (A)\}$ contains $E$, is closed under complements by subtracting from the common finite total, and under countable disjoint unions by countable additivity. It is a lambda-system containing the pi-system $\mathcal{A}$. By [F7], $\mathcal{S}\subseteq \mathcal{D}$. This also covers zero total mass and $E=\mathrm{\varnothing }$; finiteness prevents subtraction of infinite totals.