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In my lecture notes of probability course I found two different notations involving $d,\mu$ and $x$: is there any difference between $\mu(dx)$ and $d\mu(x)$?

For example I read $\mu(dx) = \frac{1}{\sqrt{2\pi}}\exp\{-\frac{1}{2}x^2\}dx$ (the density of $\mu=\mathcal{N}(0,1)$, the standard normal distribution) and $\varphi(t) = \mathbb{E}\left[\exp\{itX\}\right] = \int \exp\{itx\}\mu_X(dx)$ but also $\int f(x)d\mu(x)$:

Is it the same if I write down $\int f(x)d\mu(x)$ or $\int f(x) \mu(dx)$ ?

S.Mond
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    They are exactly the same thing. I think that the most common is $d\mu(x)$, but I saw $\mu(dx)$ used e.g. in Schilling's "Measures, Integrals and Martingales" –  Jun 09 '16 at 08:09
  • in my opinion : $\mu(dx)$ is consistent with the definition of a measure $\mu([a,b]) = \ldots$. while $d\mu$ is consistent with the Lebesgue integral $\mu([a,b]) = \int_a^b 1 d\mu$, and $d\mu(x)$ is a notation artefact for indicating the integration variable in a Lebesgue integral, or is an alias for $d\mu([0,x])$ (being consistent with the Riemann integral) – reuns Sep 14 '16 at 09:20
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    The question was already nicely answered here: https://math.stackexchange.com/a/45160/238307 – wueb Aug 10 '20 at 19:21
  • @user258700 By what standard you think you measured which one is used more often? It might differ between communities, but as pointed out in the thread linked above especially for people working in probability theory the latter is seen way more often. – wueb Aug 10 '20 at 19:25

1 Answers1

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For the sake of giving this question a definite answer:

They mean exactly the same thing. Just two different notational conventions.

Willie Wong
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