What is the meaning of $g(dx)$ or $dg(x)$ in an integral

Question

How can I interpret/solve an integration like:

$$ \int f(x) \mathbf{d}g(x) $$

or

$$ \int f(x) g(\mathbf{d}x) $$

Answer 1

Depending on the type of integration you are doing, the first integral is a Riemann-Stieltjes or a Lebesgue-Stieltjes integral.

I'll assume you're looking at integration in the Riemann sense (hence, a Riemann-Stieltjes integral).

Under certain conditions (namely, that $g(x)$ is differentiable), you can write $$\int f(x) dg(x) = \int f(x)g'(x)dx.$$

This may not always be the case. In fact, the integral exists even when $g(x)$ and $f(x)$ have countably many discontinuities -- as long as those discontinuities are not in the same place!

One way of looking at the Riemann-Stieltjes integral is thus:

For a normal Riemann integral, you are computing the area under a curve along some interval in the $x$ axis. This $x$-axis could represent, say, position.

In a Riemann-Stieltjes integral, you are integrating over an interval that is being transformed by some monotonic function $g$. So imagine that you are integrating over a distance being warped by a theoretical warp drive.

More generally, this integral is useful when you need to integrate over some set that has a lot of "weirdness" -- this arises in probability theory, for instance.

---

Regarding your second integral, and with some enlightenment from the book you linked, this is an example of a Lebesgue Integral. Lebesgue integration is similar to Riemann integration, but, loosely speaking, it uses the concept of "measure" as the thing that we integrate over.

In this case, the author constructs a measure based on an indicator function (essentially, this creates a measure equal to the number of times something happens). Then, the notation in your second integral indicates we're computing a Lebesgue integral over this measure.

Incidentally, if you're unfamiliar with Lebesgue integration, the statement on Page 3 of your book might be more enlightening if you read it right to left.