What is the meaning of $d\mu$? I'm reading measure theory and cannot find any definition of $d\mu$. The meaning of the measure $\mu$ over sets is always defined in all sources, but not that of $d\mu$. Why is the meaning of $d\mu$? Why is $d\mu=1$ for counting measure?
 A: The symbol $\mathrm d\mu$ only appears within integrals, as in $\int_Af\,\mathrm d\mu$. What it means is that we are integrating with respect to the measure $\mu$. And, by definition, saying that $\mu$ is the counting measure means $\mu(A)=\#A$; in particular, $\mu\bigl(\{x\}\bigr)=1$, for each $x$.
A: The notations $\int_a^bf(x)\,\mathrm d\mu$  or $\int_a^bf(x)\,\mathrm d\mu(x)$ are (IMHO) not very helpful. A perhaps more informative equivalent notation is$$\int_a^bf(x)\mu(\mathrm dx).$$Let us think (rather coarsely) of $\mathrm dx$ as representing a small division of the interval $[a\;\pmb,\; b]$ about the point x, say  $[x-\frac12\varepsilon\;\pmb,\; x+\frac12\varepsilon)$ of length $\varepsilon$. Then, in the simple case of the Riemann integral, the (inexplicit) measure applied to this interval is just its length $\varepsilon$, so that $\mu(\mathrm dx)=\varepsilon$. In the Riemannian case, which preceded the idea of measure, the "$\mathrm dx$" represents the (infinitesimal) length $\varepsilon$ of the interval, rather than the interval itself.
