What is the meaning of making the differential of an integral the argument of a function?

Asked Feb 10 '21 at 17:02

Active Feb 10 '21 at 17:02

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I've seen a notation in a paper amounting to $$\int f(x) g(dx).$$

That is, the differential is used as the argument of a function.

My understanding is that Riemann integrals do not allow such notation. I assume this is informal shorthand for a more standard type of integral. What is its meaning?

One guess for such meaning is to apply a variable transformation $y = g(x)$ and have the above equivalent to $$\int f(g^{-1}(y))dy,$$

Is that correct?

PS: this seems very related to What is the meaning of writing the differential inside of a function? but both that question and answer seem quite more complicated than what I am asking here.

asked Feb 10 '21 at 17:02

user118967

Note: It's a generalization of the Riemann integral called the Riemann-Stieltjes integral – user170231 Feb 10 '21 at 17:05
1

Stieltjes integral was my first thought too, but it doesn't look right. For those it is supposed to be dG(x) or some such, I thought. I can't tell what this is supposed to mean. – RobertTheTutor Feb 10 '21 at 17:08
That is not the notation of a Riemann-Stieltjes integral. – Randall Feb 10 '21 at 19:03

What is the meaning of making the differential of an integral the argument of a function?

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