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I see integrals, particularly in theoretical physics, such as $$\int f d\mu$$ where they claim $d\mu$ is some measure (often satisfying some physical properties).

I am used to seeing this notation as shorthand for either the Radon-Nikodym derivative, or defining a measure in terms of another (e.g. $dx = f(g)dy$). However, in situations where there is no second measure used as a reference, why are differentials used when defining or talking about a measure?

EDIT: My apologies, I must have phrased my question poorly. My question is why is the measure $\mu$ referred to as $d\mu$ outside the context of an integral, why is it not referred to as just $\mu$? Does the notation $d\mu$ mean anything different than just $\mu$?

CBBAM
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    This question is unclear to me. That notation is the notation for any Lebesgue integral of any “type”, so long as the (signed)(complex) measure $\mu$ is defined, along with the measure space along which $f$ is integrated. If you have a specific context involving, say, this “$g$”, it would be good to show it – FShrike Aug 17 '22 at 17:53
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    The notation $\int f,d\mu$ is common in measure theory. No one claims that $d\mu$ is a differential. It is notation for the limit $\sup_n\int f_n,d\mu$ where $f_n$ are simple functions such as $1_A$ and $\int 1_Ad\mu$ is nothing else than $\mu(A)$. – Kurt G. Aug 17 '22 at 17:55
  • @FShrike Sorry I should have specified, my question pertains to when $d\mu$ is referenced outside an integral. Why is it not referred to as simply $\mu$? – CBBAM Aug 17 '22 at 17:58
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    @KurtG. I was referring to when the measure $d\mu$ is referred to outside of an integral, why is it not referred to as simply $\mu$? Does $d\mu$ mean something different than simply $\mu$? – CBBAM Aug 17 '22 at 17:59
  • Outside the integral it is in fact often just plain $\mu$. Either convention works. – Ethan Bolker Aug 17 '22 at 18:02
  • @EthanBolker I see, so it is merely notational? – CBBAM Aug 17 '22 at 18:02
  • In case you mean something else than $d\mu = p(x)dx$ where $p(x)$ is $\mu$'s density w.r.t. to Lebesgue measure you should probably give a reference. This $d\mu$ notation is very intuitive to me. It is in fact related to the Radon-Nikodym density. – Kurt G. Aug 17 '22 at 18:03
  • @CBBAM Yes, that's all. – Ethan Bolker Aug 17 '22 at 18:15

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The notation $d\mu$ outside an integral means the same that $\mu$, however the first is used to compare with another measure, that is, when we write things like $d\mu=fd\nu$. This is just a notational costume coming from the fact that the notation $f\nu$ seems too weird to denote $\mu$ as we already have the expression $fd\nu$ inside an integral to denote the measure.

  • What if it is given without reference to another measure? – CBBAM Aug 17 '22 at 23:49
  • @CBBAM I never see that. Anyway you knows that the notation $d\mu$ just refers to the notation $\mu$ –  Aug 18 '22 at 05:45