Reading a paper concerning transport of measures, I came accross the following sentence:
"...the fundamental fact that a generic measure $\mu$ can be written as the superposition of elementary Dirac masses, that is: $$ \mu = \int_{\text{supp }\mu} \delta_x \, d\mu (x), $$ where $\text{supp }\mu$ belongs to an appropriate $\sigma$-algebra and the representation has to be understood in the sense of Bochner integrals. "
I conviced myself that this is reasonably true working with toy examples involving step functions (i.e, a finite number of Dirac measures in principle), but I cannot find a reference for the general case which - at least according to the authors - should be a well-known fact; neither I can find a proof on my own since the definition of Bochner integral is quite abstract. Any kind of help is very much appreciated.