A randomized procedure has a measurable action law

Fix $\theta \in \mathrm{\Theta }$. For each $B\in \mathcal{G}$, the function $x\mapsto K(x,B)$ is measurable and lies in $[0,1]$ by [F2]. Thus its integral is finite and at most $\int 1d{P}_{\theta }=1$ by [F1], [F3], [F8], and [F12]. Since $K(x,\mathrm{\varnothing })=0$ and $K(x,\mathsf{D})=1$, the simple-integral formula and agreement [F5, F6] give $Q_{\theta }(\mathrm{\varnothing })=0$ and $Q_{\theta }(\mathsf{D})=P_{\theta }(\mathcal{X})=1$.

By [F9], choose nonnegative simple functions $g_n\uparrow g$. Fix $n$ and write $g_n={\sum }_{j=1}^m{c}_j{1}_{B_j}$ with finite $m$, $c_j\ge 0$, and pairwise disjoint $B_j\in \mathcal{G}$, as in [F4]. The simple-integral formula and agreement [F5, F6] give $h_{g_n}(x)={\sum }_{j=1}^m{c}_{j}K(x,B_j)$. Each summand is measurable by [F2], and finite sums and nonnegative scalar multiples are measurable by [F13]. For every $x$, monotone convergence under the probability measure $K(x,\cdot )$ gives $h_{g_n}(x)\uparrow h_g(x)$. Thus $h_g$ is measurable by the increasing-limit clause of [F13].

If $(B_j{)}_{j\in \mathbb{N}}$ are pairwise disjoint members of $\mathcal{G}$, then for every $x$, $K(x,{\bigcup }_j{B}_j)={\lim }_{N\to \mathrm{\infty }}{S}_N(x)$, where $S_N(x)={\sum }_{j=0}^{N}K(x,B_j)$, by countable additivity of $K(x,\cdot )$. Each $S_N$ is measurable by [F2] and finite-sum closure [F13], and $S_N$ increases. Monotone convergence [F10] and finite additivity of the nonnegative integral [F7] give $Q_{\theta }({\bigcup }_j{B}_j)={\lim }_N{\sum }_{j=0}^N{Q}_{\theta }(B_j)={\sum }_j{Q}_{\theta }(B_j)$, where the last expression is the nonnegative extended sum [F11]. Together with step 1.1 and [F12], this proves that $Q_{\theta }$ is a probability measure.

For an indicator $g=1_B$, [F5] and [F6] give $h_g(x)=K(x,B)$ and $\int gd{Q}_{\theta }=Q_{\theta }(B)$, so the defining equation for $Q_{\theta }$ proves the iterated-integral identity. Finite addition [F7] and nonnegative homogeneity [F8] extend this equality from indicators to every nonnegative simple $g$.

The increasing simple sequence from step 1.2 satisfies $g_n\uparrow g$ and $h_{g_n}\uparrow h_g$. By step 1.2 all $h_{g_n}$ and $h_g$ are measurable; by step 2.1, $Q_{\theta }$ is a probability measure. Applying monotone convergence [F10] under $Q_{\theta }$ and under $P_{\theta }$ to the simple identities of step 3.1 proves the asserted equality, including when its value is $+\mathrm{\infty }$.