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 $\langle \mathbf 1_A, \mu\rangle$.
Another way of saying this is perhaps a measure can be thought of as a generalised function (and nothing more), much like a distribution can be thought of as one.
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.
Are there any disadvantages of this notation?
I like your proposal. Here is another context that you might consider.
With differential forms we distinguish between functions and forms. The $d$ operator takes a function and produces a form. Both of these notations have meaning. If $\mu : \mathbb{R}\to\mathbb{R}$ is a smooth function, then $d\mu$ is a form, and $\mu(x)dx$ is a form. Note that these are two different forms. We do have an equation $d\mu = \mu'(x) dx$ between forms (note the derivative).
If $P(x)$ is the cumulative density function (CDF) of a probability measure, then the appropriate notation with differential forms is $E[f] = \int f(x) dP$. If $p(x) = P'(x)$ is the probability density function (PDF), then the appropriate notation is $E[f] = \int f(x) p(x) dx$.
So if we view a measure $\mu$ as a CDF-like object, then $\int f(x)d\mu$ would be the appropriate notation, but if we view $\mu$ as a PDF-like object, then $\int f(x)\mu(x)dx$ would be the appropriate notation.
One more or less objective way to evaluate the merits of notation is to evaluate in which cases the notation can be interpreted literally. With differential forms we can interpret both of these notations literally in certain cases. The classical notation works literally if $\mu : \mathbb{R}\to\mathbb{R}$ is a CDF-like function. Your proposed notation works literally if $\mu : \mathbb{R}^n\to\mathbb{R}$ is a PDF-like function. Hence you could argue that your notation is better, since it works literally in the more general (multidimensional) case, whereas the classical notation does not. Secondly, PDFs are arguably more natural than CDFs, and make a lot of sense even when $\mu$ is a distribution.
Another possibility is to clearly distinguish between measures and functions and make no attempt to use notation to blur the difference. Then we must use $\int f \mu$, and if $\mu = dx$ we have the Lebesgue measure. Although this is the most unusual choice, this is perhaps the best from a theoretical point of view. If $\mu$ is a measure on $\mathbb{R}$, then we can write $\frac{\mu}{dx}$ for the PDF-like function that we get out the Radon-Nykodym theorem, where $dx$ is the Lebesgue measure. Conversely, if $p$ is a PDF-like function we can create the measure $p dx$. If P is a CDF-like function we can create the measure $dP$. Note that $dx$ is a special case of this, where $x$ is the identity function. This notation is perhaps the most versatile, the most concise, the most theoretically consistent, and agrees with differential forms notation. But it is also the most unusual.