Feynman–Kac path Radon–Nikodym weight

For each $0\le s\le t$, let $P_s^M$ and $Q_s$ be the target and proposal finite-prefix path laws supplied by [F3]. Initially choose a real-valued Radon–Nikodym representative ${\tilde{\rho }}_0$ from [F4]. Since ${\mu }_0$ is positive, for $N_{-}:=\{{\tilde{\rho }}_0<0\}$ its setwise identity gives $0\le {\mu }_0(N_{-})={\int }_{N_{-}}{\tilde{\rho }}_{0}d{\nu }_0\le 0$. Thus ${\int }_{N_{-}}{\tilde{\rho }}_0^{-}d{\nu }_0=0$, and [F9] implies ${\tilde{\rho }}_0^{-}=0$ ${\nu }_0$-almost everywhere. Replace ${\tilde{\rho }}_0$ by the measurable nonnegative version ${\rho }_0=\max ({\tilde{\rho }}_0,0)$; [F10] gives measurability and the modification preserves the setwise density identity. Hence $P_0^M\ll Q_0$ with base density ${\rho }_0$; after weighting by $G_0$ in [F1], ${\gamma }_0$ has density $W_0=G_0{\rho }_0$.

First record the nonnegative extension used below. If finite measures $P,Q$ satisfy $P(A)={\int }_{A}udQ$ for every measurable $A$, with $u\ge 0$, choose by [F7] simple functions $g_n\uparrow g$ for any nonnegative measurable $g$. The simple-function identity in [F5] gives $\int g_{n}dP=\int g_{n}udQ$ for each $n$; [F8] and monotone convergence of $g_{n}u$ give $\int gdP=\int gudQ$, with infinity allowed. All measures to which this fact is applied below are probabilities, so the finite-level-set condition in [F5] holds.

Fix $1\le k\le t$ and assume $P_{k-1}^M\ll Q_{k-1}$ with density $D_{k-1}(x_{0:k-1})={\rho }_0(x_0){\prod }_{j=1}^{k-1}{r}_j(x_{j-1},x_j)$. For nonnegative product-measurable $h$, define $\psi (x_{0:k-1})=\int h(x_{0:k-1},x_k)M_k(x_{k-1},d{x}_k)$; [F6] makes $\psi$ measurable. For $Q_{k-1}$-almost every history, the kernel Radon–Nikodym identity and step 1.2 give $\psi (x_{0:k-1})=\int h(x_{0:k-1},x_k)r_k(x_{k-1},x_k)R_k(x_{k-1},d{x}_k)$. Step 1.2 also extends the induction density identity from sets to $\psi$, so ordered integration from [F3] gives $\begin{array}{rl}{\displaystyle \int hd{P}_k^M}& {\displaystyle =\int \psi d{P}_{k-1}^M}\\ {\displaystyle }& {\displaystyle =\int D_{k-1}(x_{0:k-1})\psi (x_{0:k-1})Q_{k-1}(d{x}_{0:k-1})}\\ {\displaystyle }& {\displaystyle =\int h(x_{0:k})D_{k-1}(x_{0:k-1})r_k(x_{k-1},x_k)Q_k(d{x}_{0:k})\mathrm{.}}\end{array}$ Thus $P_k^M\ll Q_k$ with density $D_k=D_{k-1}{r}_k$. The kernel identity is needed only on a $Q_{k-1}$-conull set; the same is true for $P_{k-1}^M$ by the induction density, and changes on the exceptional set do not affect these nonnegative integrals.

Induction in step 2.1 gives the proposal density of every finite-horizon state path. The target measure ${\gamma }_s$ is obtained by integrating the nonnegative path potential $V_s={\prod }_{j=0}^s{G}_j$ against $P_s^M$. For each measurable $A$, apply step 1.2 to $g={\mathbf{1}}_A{V}_s$ and the density $D_s={\rho }_0{\prod }_{j=1}^s{r}_j$ to obtain ${\gamma }_s(A)={\int }_A{V}_s{D}_{s}d{Q}_s={\int }_A{W}_{s}d{Q}_s\mathrm{.}$ Dividing by $Z_s$ is valid exactly when $0<Z_s<\mathrm{\infty }$.