The standard notation for integrating with respect to a measure $\mu$ is:
$$\int f(x)\,d\mu(x).$$
But I've wondered if it could be better written as:
$$\int f(x)\mu(x)\,dx$$
where $\mu(x)$ is now thought of as a density. Then applying the measure $\mu$ to a set $A$ can be expressed as $\int \mathbf 1_A(x) \mu(x)\,dx$ or $\langle \mathbf 1_A, \mu\rangle$.
In particular, if $\mu$ is the Dirac $\delta$ measure, then integrating with respect to $\delta$ can be written $\int f(x)\delta(x)\,dx$ instead of the more awkward $\int f(x)\,d\delta(x)$. If $\mu$ is the Lebesgue measure, then we can denote it as $1$, and write $\int f(x)\cdot 1\,dx$ for the integral of $f$ with respect to $1$. If $\mu$ is a probability measure $p$, then we can write $\int f(x)p(x)\,dx$ for $\mathbb E_p[f(X)]$.
Regarding Stieltjes measures, the appropriate notation for the Stieltjes measure of a monotonic right-continuous function $g$ is $g'$, not $dg$.
I got this idea from reading the nLab entry on the Radon-Nikodym Theorem. There, it's pointed out that the Radon-Nikodym "derivative" of $\nu$ with respect to $\mu$ can be written as $\frac\nu\mu$. Notice that when written this way, it doesn't actually look like a derivative. So perhaps, the terminology "Radon-Nikodym derivative" is misleading. Besides, if it were really useful to see it as a derivative, then surely it would satisfy the product rule, but it doesn't.
Are there any disadvantages of this notation?
