I'm reading a signal processing textbook and they frequently write out formulas where if I have a Stieltjes integral:
$$\int_a^b f(t) \, dX(t)$$
They'll write an explicit formula for $dX(t)$, such as
$$dX(t) = g(t)$$
Is the best way to interpret this notation to read my integral as the limit of the following sum?
$$\sum f(t^*) (g(t_i) - g(t_{i-1}) )$$
The notation that I would expect to see would be to write $X(t) = g(t)$ and then interpret the integral as the limit of the above sum.