In Shreve's Stochastic Calculus in Finance, the Markov property is defined as
Definition 2.3.6. Let $(\Omega,\mathcal F,P)$ be a probability space, let $T$ be a fixed positive number, and let $\mathcal F(t), 0 \leq t \leq T$, be a filtration of sub-$\sigma$-algebras of $\mathcal F$. Consider an adapted stochastic process $X(t), 0 \leq t \leq T$. Assume that for all $0 \leq s \leq t \leq T$ and for every nonnegative, Borel-measurable function $f$, there is another Borel-measurable function $g$ such that $$E(f(X(t))| \mathcal F(s))=g(X(s)). (2.3.29)$$ Then we say that the $X$ is a Markov process.
That is different from the Markov property I learned before, which was defined directly for the conditional distributions of a stochastic process: the conditional distribution of future state given the past and present states is equal to the conditional distribution of the future state given the past.
Are Shreve's definition and the one I learned before equivalent, because of Riesz representation theorem which states the equivalence between measures and integrals relative to the measures?
Is it always possible to apply Riesz representation theorem to rephrase statements about measures equivalently into statements about their integrals?
In Shreve's book, $g$ is said to be dependent on $s$, but I think it also depends on $t$ because the RHS of the equation depends on both $s$ and $t$. Am I correct?
Thanks!