I have seen the following three notations and was wondering if they were all equivalent:
Over $(S, \mathcal{A}, \mu)$,
$\int_S f \, d\mu= \; \int_S f(x) d\mu(x) = \int_S f(x) \mu(dx)$
to denote the act of integrating over a measure? Moreover, while (at least in my course), we assume this to be the Lebesgue integral over the bounds of S which can, should it Riemann integrable over these same bounds, be rexpressed as a Riemann integral - is this true in the general case?
Yes, the notations are all equivalent, the last one (I believe) being more prominent in probability theory. In the real line, you have that if a function is Riemann integrable, then it is Lebesgue integrable and the values are the same. Meaning $$\int_a^b f(x)\,{\rm d}x = \int_{[a,b]} f(x)\,{\rm d}\mathfrak{m}(x).$$For this reason, people often use the left-hand side above to also denote Lebesgue integrals. For more details, you can check the end of Section 3 of Chapter 2 in Folland's Real Analysis book.
