Weird notation in probability book

Question

While studying some topics in probability, I came across with the following notation:

$$\int f(x)\mu \left( dx\right) $$

where $\mu$ is a probability measure.

I understand that this is the same as

$$\int f(x)d\mu \left( x\right) $$

but I'm wondering if there is a reason (possibly informal) for this kind of notation.

Thank you.

Answer 1

The first notation may be preferred in textbooks in probability as it is more reminiscent of a probability density function induced by some random variable $X$; namely $\mu_X(dx) = P(X\in dx)= f_X(x)dx$

Answer 2

We may (very loosely) think that $$\int_\Omega f(x)\mu(dx)$$ is the limit of sums of the form $$\sum_{i=1}^n f(x_i)\mu(A_i),$$ where $\{A_1,\dots,A_n\}$ is a partition of $\Omega$ into "small" sets, and $x_i \in A_i$. Following Leibniz, we write $dx$ for "a small set containing $x$".