In our class on stochastic processes I keep getting confronted with a problem w.r.t. notation. I am a mathematics major and attended all the prerequisite classes on standard analysis.
In those analysis classes we introduced the Lebesgue-Integral as a limit of sums over "simple functions" for example like this (taken from wikipedia)
$$ \int_{\Omega} f \, d\mu := \lim_{n \rightarrow \infty} \int_{\Omega} f_n \, d\mu$$
where the $f_n$ are a sequence of simple functions that converges pointwise and grows monotonously. The integral notation is introduced first for simple functions of the form $\phi := \sum_{i=1}^{n} \alpha_i \mathbb{1}_{A_i} $ like so
$$ \int_{\Omega} \phi \, d\mu := \sum_{i = 1}^{n} \alpha_i \mu(A_i) $$
I liked this introduction and found it very reasonable and intuitive. What troubles me now is that the $d\mu$ and later similarly the $dx$ have no properly defined meaning outside of the context of this definition. I realize there is another definition when differentiation is introduced, but there too we defined $\frac{df}{dx}$ via a limit and as if it was merely like a name without an intrinsic meaning.
Yet, in our class we use $dy$ e.g. in the following description of the Chapman-Kolmogorov $(*)$ condition for transition functions like this:
$$\forall s \leq t \leq u: \mu_{s,t}(x,B) = \int_{E} \mu_{s,u}(x,dy) \mu_{u,t}(y,B) $$
For context: For a stochastic process $(X_t)_{t \in T}$ on a state space $E$ the transition function $\mu_{s,t}(x,B)$ gives the probability that $X$ transitions into $B \subset E$ starting from point $x \in E$ within the timeframe $[s,t]$.
Suddenly something that used to be nothing but the indicator for the variable/measure we are integrating over is itself subjected to a function. From context I would say it represents a set (a small ball around $y$ maybe) but I don't know how to deal with it in a rigorous way. I.e. even if I was given specific values for $s,t,u,x $ and $B$, I wouldn't know how to calculate or even properly represent such an integral.
I read some articles on differentials/infinitesimals e.g. from the San José State University or on Wikipedia>Differential but I still feel clueless as to
- How to interpret this notation; e.g. what exactly am I summing up, integrating over in $(*)$?
- How to calculate with these infinitesimals/differentials in a practical way?
In your C-K example, it's a matter of sorting out the notation. You are integrating a (measurable) function $f(y):= \mu_{u,t}(y,B)$ (in which $u,t,B$ are fixed, for the moment) with respect to a measure $\nu$ defined by $\nu(A):=\mu_{s,u}(x,A)$ (in which $s,u,x$ are fixed). This is a Lebesgue integral of the type you learned in analysis and described above. What may be novel is that the function $f$ and the measure $\nu$ depend on parameters, and in the end so does the integral $\int_E f d\nu$. It can be shown that if $(s,u,x)\mapsto\mu_{s,u}(x,A)$ and $(u,t,y)\mapsto\mu_{u,y}(y,B)$ are suitably measurable, then the end result $(s,t,x)\mapsto\mu_{s,t}(x,A)$ is measurable in the same sense.