Does anyone know if the notion of a Dirac measure was named after P.A.M. Dirac? If yes, did he actually introduce it, or is it because integration w.r.t. a Dirac measure does what the so-called Dirac delta function is supposed to be used for?
Yes, the Dirac measure is so named because it performs the same action as Dirac's "delta function." I don't believe Dirac used the measure theory formalism, however - he was a physicist.
I believe Dirac Measure, Dirac distribution, and Dirac function (an entrenched misnomer) are synonyms for the "unit impulse". The unit impulse was used before P. A. M. Dirac by Cauchy, Poisson, Kirchhoff, and Helmholtz but they did not call it the Dirac measure, distribution, or function. Dirac called it the Delta function in his quantum theory work and the name stuck. So It was named after Dirac--the Dirac Delta Function.