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?