Consider the following stochastic differential equation $$ \mathrm d X_t = b(X_t) \, \mathrm d t + \mathrm d W_t, \qquad X_0 = \xi, \qquad \xi \sim \mu $$ where $(W_t)_{t \geq 0}$ is a standard Brownian motion on $\mathbb R$ and $\mu \in \mathcal P(\mathbb R)$ is a given probability distribution on $\mathbb R$ such that $\xi$ has finite second moment. Assume that $b$ is smooth and globally Lipschitz continuous, so that a unique strong solution exists, with law denoted by $\mathbb P_{\mu}$ on the space of continuous functions $C([0, \infty))$. Suppose that $\mathcal F$ is a functional on $C([0, \infty))$ that is measurable with respect to the Borel $\sigma$-algebra of $C([0, \infty))$ and integrable with respect to $\mathbb P_{\mu}$. I would like to justify rigorously that $$ \mathbb E_{\mu}[\mathcal F] = \int_{\mathbb R} \mathbb E_{\delta_x}[\mathcal F] \, \mu(\mathrm d x), $$ where $\mathbb E_{\delta_x}$ is the expectation under $\mathbb P_{\delta_x}$, which itself is the law of the solution over path space with the deterministic initial condition $X_0 = x$. What is the simplest approach to this end? It seems that the result follows from Corollary 2.2 in the classical paper by Yamada and Watanabe, but I am wondering whether there is a simpler answer.
In particular, does this follow evidently from the proofs of existence of a weak solution, with which I am not familiar?
I think the question may be related to showing that $(x, A) \mapsto \mathbb P_{\delta_x}(A)$ is a regular conditional probability of $(X_t)_{t\geq 0}$ with initial condition $\xi \sim \mu$ given $X_0 = x$, but not sure this helps.
