Q: Notation regarding integral w.r.t. a measure While reading articles about statistical inference and SMC algorithms I have come by the following notation (never seen in school) to represent the expectation w.r.t. a measure $\mu$
$$
\mathbb{E}[\phi(X)] = \int\phi(x) \mu(\text{d}x),
$$
which I have come to understand means
$$
\int\phi(x) \text{d}\mu(x), \text{ or }  \int\phi(x) \mu'(x)\text{d}x.
$$
However, when defining the empirical measure $\mu_N(\text{d}x)$
$$
\mu_N(\text{d}x) := \frac{1}{N}\sum^N_{i=1}\delta_{X^i}(\text{d}x),
$$
does the notation
$$
\text{d}\delta_{X^i}(x)
$$ 
make any sense? Or can we just use notation
$$
\delta_{X^i}(x)\text{d}x
$$
in this case?

A: I don't think I have ever seen the notations $d\delta_{X^i}(x)$ or $\delta_{X^i}(x)dx$. However the former seems correct to me, as $d\mu(x)$ is a common notation. But I don't really like the latter, as it seems to imply that there's a density w.r.t. the Lebesgue measure, which is of course wrong.
If I had to define the empirical measure, I would just write:
$$\mu_N=\frac{1}{n}\sum_{i=1}^N\delta_{X^i}$$
which perfectly makes sense and is unambiguous.
When I'm writing an integral I prefer to use the notation $\int f(x)d\mu(x)$, or simply $\int fd\mu$, rather than $\int f(x)\mu(dx)$, but both notations exist and I think it's just a matter of habit
