Let $\left\{X_t \right\}_{t\in T}$ be a time homogeneous Markov process with state space $S$. How do I formally demonstrate$$P(X_t\in B)=\int_S P(X_t\in B\mid X_0)dPX_0^{-1}$$(here $PX_0^{-1}$ is the distribution of $X_0$.)
At first the formula $\int_{g^{-1}(B)}f\circ gd\mu=\int_Bfd\mu g^{-1}$ seemed relevant, but I don't really know what to do.
Thanks in advance!
Let $\mu$ denote the distribution of $X_0$ then the formula$$P(X_t\in B)=\int_S P(X_t\in B\mid X_0)\mathrm d\mu$$ makes no sense since $P(X_t\in B\mid X_0)$ is a random variable, defined on the source set $\Omega$ of $X_0$ while $\mu$ is a measure on the target set $S$ of $X_0$. Instead, one has $$ P(X_t\in B)=\int_\Omega P(X_t\in B\mid X_0)\mathrm dP=\int_S P(X_t\in B\mid X_0=x)\mathrm d\mu(x), $$ which is a consequence of the definition of the random variable $Y=P(X_t\in B\mid X_0)$ since $Y$ is such that, for every bounded function $u$, $$ E(u(X_0)Y)=E(u(X_0);X_t\in B), $$ or, equivalently, $$ \int_\Omega u(X_0)Y\mathrm dP=\int_{\{X_t\in B\}}u(X_0)\mathrm dP. $$ In the present case, use the function $u=1$.