I just read in a book on stochastic optimization, this weird integral: $$\int V(x,z')Q(z,\mathrm dz')$$ where $z$ is a random variable and specifically a markov chain, $Q$ is the so called "state transition function" (don't know exactly what it is yet). The book says this is a Lebesgue-Stieltjes integral. But I thought I've seen Lebesgue integrals before, but they just have the standard form $\int \cdots\mathrm d\mu$ for a measure $\mu$.
I've never seen a "$\mathrm d\cdot$" inside a function in the integral. What does this mean?
$\int f(x)\mu(dx)$ is another standard (but slightly rarer) notation for $\int f(x)\,d\mu(x)$ or $\int f\,d\mu$, used mainly in probability theory. See p.177 of V.I. Bogachev's Measure Theory for a usage making this meaning clear. The notation is sometimes clearer in contexts where there are multiple measures and variables at hand. (You might find this intuitive: since $\mu$ is a measure we know about $\mu(A)$ for sets, and you can think of $\int f(x)\mu(dx)$ as being like a Riemann sum, writing the space as a union of lots of little sets $[x,x+dx]$ or $dx$, each getting weight $\mu([x,x+dx])$ or $\mu(dx)$.)
Another freaky notation is $\mu(f)$ for $\int fd\mu$, emphasizing the role of a measure as a functional, mapping functions $f$ to numbers $\mu(f)$.
In your case, $V(x,z')$ is a function of two states (numbers or points or whatever) and $Q(z,\cdot)$ is a measure, presumably the conditional distribution of your process's next state $z'$ given the current state $z$. And your integral is what you get when you integrate $V(x,z')$ over $z'$ with respect to this measure.