Consider stochastic proces $\left\{X_t\right\}_{t\in T}$, where $T=\left<0,\infty\right)$ and for each $t\in T$ $X_t$ can be continuous, dicrete or other random variable on same probability space $\left(\Omega,\mathscr{S},P\right)$.
If for each $t\in T$ $X_t$ is continous random variable and $T=\left<0,\infty\right)$, then Markov property can be defined (I think) as $$ \forall t,s\in T, \forall B\in \mathscr{B}:P(X_{t+s}\in B|\sigma(X_l; l\leq t))=P(X_{t+s}\in B|\sigma(X_t)). $$
Moreover, I think that equation does not have to hold everywhere, only $P$-almost everywhere. Because the condition probability is defined as the function which is $\mathscr{F}_0$ measurable and for all $G\in\mathscr{F}_0$ satisfy $$ \int\limits_{G}P(A|\mathscr{F}_0)dP=P(A\cap G), $$ where $A\in\mathscr{F}$, $\mathscr{F}_0$, $\mathscr{F}$ are $\sigma$-algebras and $\mathscr{F}_0\subset \mathscr{F}$. The existence of this function (of the conditional probability) is guaranteed by Radon-Nikodym theorem but only $P$-almost everywhere.
If I am correct and I use the definition of conditional probability in Markov property, I have $$ \forall t,s\in T, \forall B\in \mathscr{B}:\int\limits_{G}P(X_{t+s}\in B|\sigma(X_l; l\leq t))dP=\int\limits_{G}P(X_{t+s}\in B|\sigma(X_t))dP=P(X_{t+s}^{-1}(B)\cap G). $$
Is the $G$ from $\sigma(X_l; l\leq t)$ or $\sigma(X_t)$?
Can you recommend me some literature which deals with Markov processes in general not only about Markov chains?
Any help will be appreciated.