TITLE: Difference between $d\mu(x)$ and $\mu(dx)$
QUESTION [9 upvotes]: 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 $\int f(x) \mu(dx)$ ?

REPLY [3 votes]: For the sake of giving this question a definite answer:
They mean exactly the same thing. Just two different notational conventions.