I'm trying to figure out what the following statement refers to.
A process $X$ is markov with transitions semigroup $(K_t)_{t\geq0}$ and initial distribution $\mu$ if and only if for all $n\in\mathbb{N}_0$, for any $0=t_0<...<t_n$ and for all measurable functions $f_0,...,f_n:S\rightarrow [0,\infty)$ we have
$\mathbb{E}[f_0(X_{t_0})\cdots f_n(X_{t_n})]=\int_S{\mu(dx_0)f_0(x_0)}\cdots\int_S{K_{t_n-t_{n-1}}(x_{n-1},dx_n)f_n(x_n)}$
What is suggesting this Statement? Why it consider expectation instead to conditional expectation, and making use of the transition semigroup? I have some trouble with the integrals too. Is it to consider as a disjoint product of integrals?
Thanks for any suggestion :)