Given a distribution function $F$ how does one interpret an integral of the form $\int (.) F(dx)$?

More specifically, I've come across an exercise in a book called 'Heavy Tail Phenomena' in which this kind of integral is used.

The question is, for a distribution $F$, in which $1 - F(x) \sim x^{-\alpha}, \ \ x \rightarrow \infty$ show (potentially by integrating by parts) for $\eta \geq \alpha$ that:

$$\lim_{x\rightarrow \infty} \dfrac{\int_0^x u^{\eta} F(du)}{x^{\eta}(1 - F(x))} = \dfrac{\alpha}{\alpha + \eta}$$

I do not understand what the $F(du)$ part of the integral means and I have never come across this before. Could somebody explain this to me?

• en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration Commented Jun 21, 2017 at 17:00
• Is $d(F(u))$ the same as $F(du)$? Sorry I'm confused on this point. Commented Jun 21, 2017 at 17:05
• Yes, at least in my experience. Commented Jun 21, 2017 at 17:06
• I've never seen the notation "$F(du)$" and suspect it is meaningless. But I think "$dF(u)$" is pretty common. Are you sure it isn't just a typo?
– MPW
Commented Jun 21, 2017 at 17:16
• Possible duplicate of What does it mean to integrate with respect to the distribution function? Commented Jun 21, 2017 at 17:20

Yes, from the given context, the symbol $d(F(u))$, when written along with the sign $\int$, is the symbol $F(du)$. Writing either one does not alter any truth.
For your reference, you may want to learn some Stieltjes integration. A Stieltjes integral is a number of the form $\int_{a}^{b}fdg$, which is defined as the limit of the sum $\sum_{1}^{n}f(x^{*}_{i})[g(x_{i+1}) - g(x_{i})]$ as the partitions of $[a,b]$ go finer and finer, provided only that this limit exists.
Now it would be clear that, as long as one understands a Stieltjes integral correctly, it does not matter he writing $\int_{a}^{b}f(u)d(F(u))$ or $\int_{a}^{b}f(u)F(du)$, although the latter could be not as clear as the former at the first glance. However, context usually helps!
As a related digression, in fact a Lebesgue integral also admits two common notations: $\int_{X}f(x)d\mu(x)$ or $\int_{X}f(x)\mu(dx)$. Although writing out $(x)$ seems redundant in a simple case, it would help in clarity in cases such as integrating a function of two arguments.