I would be interesting in understanding the precise definition of the integral $$f(A)=\int_a^b p(t,A) dt \quad A \in \mathcal B(\mathbb R)$$ with $P$ a suitable function from $\mathbb R$ to the space of measures. I find this integral quite often when, for example, dealing with the Chapman Kolmogorov equation.
My question is how is this integral defined?
If we fix $A$ we could define $p_A(t)=p(t,A)$ and hence for every $A$: $$f(A)=\int_a^b p_A(t) dt $$ where the integral is the classical Riemann integral.
But, is this equivalent to define f as the measure such that $$\| f- \sum_i(\cdot,t_i)(x_{i+1}-x_i ) \|_M \rightarrow 0$$ as $C(P) \rightarrow 0$, where $P$ is a partition of the interval $[a,b]$ in intervals $[x_{i+1},x_i] \ni t_i$ and $C(P)$ its mesh and $\| \cdot \|$ is a norm in the space of measure, for example total variation.
What confuses me is that I never read the definition of such an integral and this confuses me.