From Wikipedia (note that I have modified it from for a Markov process to for a general stochastic process):

the conditional probability density $p_{i;j}(f_i\mid f_j)$ is the transition probability between the times $i>j$. So, the Chapman–Kolmogorov equation takes the form $$ p_{i_3;i_1}(f_3\mid f_1)=\int_{-\infty}^\infty p_{i_3;i_2, i_1}(f_3\mid f_2, f_1)p_{i_2;i_1}(f_2\mid f_1) \, df_2. $$

I was wondering how to rephrase the above equation in terms of conditional probability measures instead of conditional probability densities?

To give you an idea of what I want, the analogy in the unconditional case, i.e. the case for finite dimensional distributions, is as following:

From Wikipedia:

Let $$ p_{i_1,\ldots,i_n}(f_1,\ldots,f_n) $$ be the joint probability density function of the values of the random variables $f_1$ to $f_n$. Then, the Chapman–Kolmogorov equation is $$ p_{i_1,\ldots,i_{n-1}}(f_1,\ldots,f_{n-1})=\int_{-\infty}^{\infty}p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)\,df_n $$

It can be written in terms of probability measures, instead of their density functions (also from Wikipedia)

for all measurable sets $F_{i} \subseteq \mathbb{R}^{n},m \in \mathbb{N}$ $$\nu_{t_{1} \dots t_{k}} \left( F_{1} \times \dots \times F_{k} \right) = \nu_{t_{1} \dots t_{k} t_{k + 1}, \dots , t_{k+m}} \left( F_{1} \times \dots \times F_{k} \times \mathbb{R}^{n} \times \dots \times \mathbb{R}^{n} \right). $$

Thanks and regards!


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