I am looking for a reference that explains the connection between a measure $\mu$ and a distribution as described for instance here:
$$\langle F,g\rangle =\int_{\Omega} g \;d\mu$$
where $g$ is some test function and $\langle.,.\rangle$ denotes the bilinear pairing between distributions and test functions. I tried looking inside the humongous 5 volumes book by Gel'fand and Shilov but I wasn't able to find anything (although I am sure it must be there, somewhere).
So if you know where I can find this I would be glad! Thanks in advance.
Uhm... Measures induce distributions, but not the other way around. If you're working on a nice-enough space, measures represent the dual of the space of continuous functions (that are zero at infinity), but distributions represent the dual of the smooth functions (that are zero at infinity)... Or Schwartz functions... Anyway, since you can "do more" with smooth functions (like: differentiate and evaluate), there are "more" distributions than measures.