We denote with $K_{s,t}$ a stochastic kernel from measurable space $(S,\mathcal{S})$ to itself. Further we assume that $P[X_t\in A|\mathcal{F}_s]=P[X_t\in A|X_s] = K_{s,t}(X_s,A)$, where $X=(X_t)$ is a stochastic process with values in $S$. I know the following result:
Let $X_i:(\Omega,\mathcal{F})\to (S_i,\mathcal{S}_i)$ with $i=1,2$ r.v. and $F:S_1\times S_2\to [0,\infty]$ is $\mathcal{S}_1\times \mathcal{S}_2$ measurable. Look at the vector $(X_1,X_2)$ as $S_1\times S_2$ valued r.v. We assume that the distribution of $(X_1,X_2)$ on $(S_1\times S_2,\mathcal{S}_1\times \mathcal{S}_2)$ is of the form $P_1\times K$ where $P_1$ is a probability measure on $(S_1,\mathcal{S}_1)$ (the distribution of $X_1$ under $P$) and $K$ a stochastic kernel from $(S_1,\mathcal{S}_1)$ to $(S_2, \mathcal{S}_2)$. Then we have $E[F(X_1,X_2)|X_1](\omega)=\int_{S_2}F(X_1(\omega),x_2)K(X_1(\omega),dx_2)$
Now why is the following equation true:
$E[K_{t,u}(X_t,A)|\mathcal{F}_s]=\int_S K_{t,u}(y,A)K_{s,t}(X_s,dy)$
for $s\le t\le u$. It seems that we take $F=K$, however this makes no sense for me. It would be appreciated if someone could explain this equation.
math