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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.

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    What you say you would expect is closer to the truth than is what you said you read. But there's no need to use two letters, $X$ and $g$ to refer to the same function. One is enough.2012-08-16

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"$dX(t)=g(x)$" makes no sense. One can write $dX(t)=g(t)\,dt$, with $t$ rather than $x$ as the variable, and with "$dt$" after it, provided $X$ is differentiable and $X'=g$. But the Riemann-Stieltjes integral makes sense even in many cases where $X$ is not differentiable, and is the limit of the sum you wrote provided you put $X(t_i)-X(t_{i-1})$ where you had $g(t_i)-g(t_{i-1})$.

Later edit: In view of the later revision of the question, I will add this:

  • "$dX(t) = g(t)$" doesn't make sense without "$dt$", i.e. one has $dX(t)=g(t)\,dt$ provided $X'=g$. But the Riemann-Stieltjes integral can make sense even in many cases where $X$ is not differentiable.
  • You need the limit of the sum with $X(t_i)-X(t_{i-1})$, not with $g(t_i)-g(t_{i-1})$.
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    I made a mistake in the question, I'll edit it. I meant to put g(t).2012-08-16