Is there any good reason for which integrating according to a measure includes a $\mathrm d$ as in $\int f\mathrm d\mu$ ? Or is it just a manner to keep formal consistency with the traditional notation for integration ?
Among others, this notation suggests the possibility of "formal differentials of measures" like in $\mathrm d \nu = f \mathrm d\mu$ and of course this is consistent with the definition of the radon-nikodym derivative $f = \frac{\mathrm d\nu}{\mathrm d\mu}$. But precisely I'm wondering if there any good reason to speak of a "derivative" here. Broadly speaking the notion of derivation generally implies linearity and Leibnitz's product rule but here I cannot see how one could make sense of this (with $\times$ the product of measures):
$\frac{\mathrm d(\nu_1 \times \nu_2)}{\mathrm d\mu} = \nu_1 \times \frac{\mathrm d\nu_2}{\mathrm d\mu} + \frac{\mathrm d\nu_1}{\mathrm d\mu} \times \nu_2$
given that neither of the members of this equation seems defined. Are they other products on measures ?
Are the name "the radon-nikodym derivative" and the notations $f = \frac{d\nu}{d\mu}$ and $\nu(A) = \int_A f \mathrm d\mu$ better in any sense than if it would have been called "the radon-nikodym transform" and written down $f = RN_{\mu}(\nu)$ and $\nu = f \bullet \mu$ ? Or do we know similarities that suggest that there is a good reason for this?
Edit
Rethinking about it I realize that equality $\frac{\mathrm d \nu}{\mathrm d \mu} \cdot \frac{\mathrm d \mu}{\mathrm d \lambda} = \frac{\mathrm d \nu}{\mathrm d \lambda}$ suggests some algebraic behaviour not implied by my notation. However this simply implies a quotient, not a derivative. So we may speak of the "Radon Nikodym quotient" and write $\frac{\nu}{\mu} \cdot \frac{\mu}{\lambda} = \frac{\nu}{\lambda}$.
Now let us go a step further. What can we say about $\frac{\mathrm d(\nu_1 \times \nu_2)}{\mathrm d(\mu_1 \times \mu_2)}$ (with $\nu_1 \ll \mu_1$ and $\nu_2 \ll \mu_2$)? My guess is that we have: $$\frac{\mathrm d(\nu_1 \times \nu_2)}{\mathrm d(\mu_1 \times \mu_2)} = \frac{\mathrm d\nu_1}{\mathrm d\mu_1} \otimes \frac{\mathrm d\nu_2}{\mathrm d\mu_2}$$ which strongly supports the quotient notation: $$\frac{\nu_1 \times \nu_2}{\mu_1 \times \mu_2} = \frac{\nu_1}{\mu_1} \otimes \frac{\nu_2}{\mu_2}$$ In the end I really feel $\int$ and $\mathrm d$ where kept for historical reasons. And that is quite unfortunate because it seems to be a false good idea.