I am studying characteristic functions in probability theory and I am struggling to understand the following equality.
Why is this transformation true?
Wikipedia states that $F_X$ is the cumulative distribution function of $X$, and the integral is of the Riemann–Stieltjes kind. But we haven't learned that yet. How do I have to understand this equity without using Riemann–Stieltjes?
What I also don't understand is what the true meaning of integrating by $dF(x)$ is.
Please explain that in terms of someone who has only visited introductions to Measure Theory, Lebesgue Integrals, and Probability Theory.
Thank you very much for your efforts!