TITLE: Is there any difference between the notations $\int f(x)d\mu(x)$ and $\int f(x) \mu(dx)$?
QUESTION [32 upvotes]: Suppose $\mu$ is a measure. Is there any difference in meaning between the notation
$$\int f(x)d\mu(x)$$
and the notation
$$\int f(x) \mu(dx)$$?

REPLY [10 votes]: At times, I find the $\mu(dx)$ notation to be quite intuitive. Informally, if we think of $dx$ as representing an infinitesimally small "chunk" of the real line, then $\mu(dx)$ is its measure.
For a formal example, let $F$ be right-continuous and increasing and $f$ continuous. Let $\mu$ be the Lebesgue-Stieltjes measure associated to $F$, that is, $\mu((a,b])=F(b)-F(a)$. Let $\{x_j\}_{j=1}^n$ be a partition of some interval $I$ and let $\Delta x_j = (x_{j-1},x_j]$. Although it is customary to let $\Delta x_j$ denote the length of this interval, in cases where we may apply different notions of length to the same interval, it may make more sense to simply let $\Delta x_j$ denote the interval itself.
In this case, we have
$$
\int_I f(x)\,\mu(dx) = \lim_{n\to\infty}\sum_{j=1}^n f(x_j)\mu(\Delta x_j),
$$
provided the mesh of the partition tends to zero. In this setting, the $\mu(dx)$ notation keeps both sides of the equality notationally consistent with one another.
That said, though, whether you choose to use $\mu(dx)$ or $d\mu(x)$, the meaning is the same.