A finite label has a conditional probability function

For each $j$, choose a version $Z_j$ of $\mathbb{E}[{\mathbf{1}}_{\{Y=y_j\}}\mid \mathcal{G}]$, which exists by [A1,F1]. Order preservation and preservation of constants give $0\le Z_j\le 1$ almost surely by [F2]. Set $N_j=\{Z_j<0\}\cup \{Z_j>1\}\in \mathcal{G}$ and define $Z_j^{\mathrm{\prime }}$ to equal $Z_j$ on $N_j^c$ and $0$ on $N_j$. Since both pieces are $\mathcal{G}$-measurable, $Z_j^{\mathrm{\prime }}$ is $\mathcal{G}$-measurable; it takes values in $[0,1]$ everywhere and equals $Z_j$ almost surely. By [F3], it has the same event integrals, so it remains a conditional-expectation version. Linearity shows that ${\sum }_j{Z}_j$ is a version of the conditional expectation of ${\sum }_j{\mathbf{1}}_{\{Y=y_j\}}=1$; the constant 1 is another version, so [F6] gives ${\sum }_j{Z}_j=1$ almost surely. Hence ${\sum }_j{Z}_j^{\mathrm{\prime }}=1$ almost surely.

For each label $j$ and rational $q\in \mathbb{Q}\cap [0,1]$, the event $\{Z_j^{\mathrm{\prime }}<q\}\in \mathcal{G}$ equals $X^{-1}(A_{j,q})$ for some $A_{j,q}\in {\mathrm{\Sigma }}_X$ by [F5]. AC permits choosing these sets for the countable family of label-rational pairs. Define ${\tilde{\eta }}_j(x)=\inf (\{q\in \mathbb{Q}\cap [0,1]:x\in A_{j,q}\}\cup \{1\})$. For $0<t\le 1$, $\{x:{\tilde{\eta }}_j(x)<t\}={\bigcup }_{q<t}{A}_{j,q}$; for $t\le 0$ this sublevel set is empty and for $t>1$ it is all of $\mathsf{X}$. Thus each ${\tilde{\eta }}_j$ is measurable and takes values in $[0,1]$ by [F4,F7].

For every $j$ and $\omega$, the rationals $q$ satisfying $X(\omega )\in A_{j,q}$ are exactly those with $Z_j^{\mathrm{\prime }}(\omega )<q$. Their infimum is $Z_j^{\mathrm{\prime }}(\omega )$ when this value is below $1$; at value $1$ the rational set is empty and the filler $1$ gives the same value. Thus ${\tilde{\eta }}_j(X)=Z_j^{\mathrm{\prime }}$ everywhere. The measurable set $D=\{x:{\sum }_j{\tilde{\eta }}_j(x)=1\}$ is well-defined by [F7]. Define $\eta (x)=({\tilde{\eta }}_1(x),\dots ,{\tilde{\eta }}_K(x))$ on $D$ and $\eta (x)=(1,0,\dots ,0)$ off $D$. Its coordinates are measurable. The inverse image of an open rational box in ${\mathbb{R}}^K$ is a finite intersection of coordinate preimages, and these boxes form a countable basis, so $\eta$ is Borel measurable into ${\mathbb{R}}^K$ and hence measurable into the subspace ${\mathrm{\Delta }}_K$. It takes values in ${\mathrm{\Delta }}_K$ everywhere, and ${\eta }_j(X)=Z_j^{\mathrm{\prime }}$ almost surely because ${\sum }_j{Z}_j^{\mathrm{\prime }}=1$ almost surely.

For every $A\in {\mathrm{\Sigma }}_X$ and label index $j$, the event $\{X\in A\}$ belongs to $\mathcal{G}$. The conditional-expectation identity [F1] and almost-sure equality [F3] therefore give $P(X\in A,Y=y_j)=\mathbb{E}[{\mathbf{1}}_{\{X\in A\}}{Z}_j]=\mathbb{E}[{\mathbf{1}}_{\{X\in A\}}{Z}_j^{\mathrm{\prime }}]=\mathbb{E}[{\mathbf{1}}_{\{X\in A\}}{\eta }_j(X)]\mathrm{.}$ This is the claimed event identity for every class.