[proofplan] We first check that the extended density is a genuine probability target: nonnegativity gives positivity of the density, and unbiasedness gives the same normalizing constant $Z$ as the original target. The theorem statement assumes that the Metropolis-Hastings transition kernel has this extended probability measure as its invariant target, so no separate detailed-balance verification is needed here. Finally, we compute the $X$-marginal of $\widetilde\pi$ by integrating out the auxiliary variable $u$ and use the unbiasedness identity to recover exactly the normalized measure $\pi$. [/proofplan]

[step:Verify that the extended target is a probability measure]Because $\widehat\gamma:X\times S\to[0,\infty)$ is measurable and nonnegative, it defines a nonnegative density with respect to the product measure $\lambda\otimes\rho$. Its total mass is \begin{align*} \int_{X\times S}\widehat\gamma(x,u)\,d(\lambda\otimes\rho)(x,u)=\int_X\left(\int_S\widehat\gamma(x,u)\,d\rho(u)\right)d\lambda(x). \end{align*} The equality follows from Tonelli's theorem applied to the nonnegative measurable function $\widehat\gamma$: the [measure space](/page/Measure%20Space) $(X,\mathcal A,\lambda)$ is sigma-finite by hypothesis, and $(S,\mathcal S,\rho)$ is sigma-finite because $\rho$ is a probability measure. By the unbiasedness hypothesis, \begin{align*} \int_X\left(\int_S\widehat\gamma(x,u)\,d\rho(u)\right)d\lambda(x)=\int_X\gamma(x)\,d\lambda(x)=Z. \end{align*} Since $0<Z<\infty$, the formula \begin{align*} \widetilde\pi(B)=\frac{1}{Z}\int_B\widehat\gamma(x,u)\,d(\lambda\otimes\rho)(x,u) \end{align*} defines a probability measure on $(X\times S,\mathcal A\otimes\mathcal S)$.[/step]

[guided]The extended Metropolis-Hastings algorithm is only meaningful if the proposed extended target is a genuine probability measure. The nonnegativity assumption is exactly what ensures that $\widehat\gamma$ can be used as an unnormalized density. Since $\widehat\gamma:X\times S\to[0,\infty)$ is measurable, the set function \begin{align*} B\mapsto \int_B\widehat\gamma(x,u)\,d(\lambda\otimes\rho)(x,u) \end{align*} is a measure on $(X\times S,\mathcal A\otimes\mathcal S)$. We must compute its total mass. Because $\widehat\gamma$ is nonnegative, Tonelli's theorem applies without any prior integrability assumption once the measure-space hypotheses are checked. The space $(X,\mathcal A,\lambda)$ is sigma-finite by hypothesis, and $(S,\mathcal S,\rho)$ is sigma-finite because $\rho(S)=1$. Therefore Tonelli's theorem gives \begin{align*} \int_{X\times S}\widehat\gamma(x,u)\,d(\lambda\otimes\rho)(x,u)=\int_X\left(\int_S\widehat\gamma(x,u)\,d\rho(u)\right)d\lambda(x). \end{align*} The inner integral is exactly the expectation of the estimator at fixed $x$, taken with respect to the auxiliary law $\rho$. The unbiasedness hypothesis states that this inner integral equals $\gamma(x)$ for every $x\in X$. Therefore \begin{align*} \int_{X\times S}\widehat\gamma(x,u)\,d(\lambda\otimes\rho)(x,u)=\int_X\gamma(x)\,d\lambda(x)=Z. \end{align*} The assumption $0<Z<\infty$ now gives both finiteness and nonzero total mass, so division by $Z$ normalizes the extended measure. Hence $\widetilde\pi$ is a probability measure.[/guided]

[step:Use the assumed invariant target on the extended space] Let $K$ denote the Metropolis-Hastings transition kernel on $(X\times S,\mathcal A\otimes\mathcal S)$ for which $\widetilde\pi$ is invariant. This means precisely that, for every $B\in\mathcal A\otimes\mathcal S$, \begin{align*} \int_{X\times S}K((x,u),B)\,d\widetilde\pi(x,u)=\widetilde\pi(B). \end{align*} Thus $\widetilde\pi$ is an invariant probability measure for the extended transition kernel $K$. [/step]

[step:Compute the marginal of the extended target] Let $\operatorname{pr}_X:X\times S\to X$ be the coordinate projection map defined by $\operatorname{pr}_X(x,u)=x$. The $X$-marginal of $\widetilde\pi$ is the probability measure $\widetilde\pi_X$ on $(X,\mathcal A)$ defined by \begin{align*} \widetilde\pi_X(A):=\widetilde\pi(A\times S) \end{align*} for every $A\in\mathcal A$. For such $A$, the restricted function $\widehat\gamma\mathbb{1}_{A\times S}:X\times S\to[0,\infty)$ is measurable and nonnegative. Since $(X,\mathcal A,\lambda)$ is sigma-finite and $(S,\mathcal S,\rho)$ is sigma-finite, Tonelli's theorem applies to this restricted function. Hence Tonelli's theorem and the unbiasedness hypothesis give \begin{align*} \widetilde\pi_X(A)=\frac{1}{Z}\int_{A\times S}\widehat\gamma(x,u)\,d(\lambda\otimes\rho)(x,u). \end{align*} Thus \begin{align*} \widetilde\pi_X(A)=\frac{1}{Z}\int_A\left(\int_S\widehat\gamma(x,u)\,d\rho(u)\right)d\lambda(x). \end{align*} Using $\int_S\widehat\gamma(x,u)\,d\rho(u)=\gamma(x)$, we obtain \begin{align*} \widetilde\pi_X(A)=\frac{1}{Z}\int_A\gamma(x)\,d\lambda(x)=\pi(A). \end{align*} Since this holds for every $A\in\mathcal A$, the $X$-marginal of $\widetilde\pi$ is exactly $\pi$. [/step]

[step:Transfer stationarity to the desired coordinate marginal] Because $K$ is invariant with respect to $\widetilde\pi$, a Markov chain on $X\times S$ initialized with law $\widetilde\pi$ has law $\widetilde\pi$ after one transition, and hence after every transition. Applying the projection map $\operatorname{pr}_X$ to this stationary extended chain gives an $X$-valued coordinate process whose one-time marginal distribution at every time is the pushforward measure $(\operatorname{pr}_X)_\#\widetilde\pi$. By the preceding step, this pushforward measure is $\widetilde\pi_X=\pi$. Therefore the $x$-coordinate marginal distribution of the stationary pseudo-marginal Metropolis-Hastings chain is $\pi$. [/step]