Of course the Dirac delta is not a function. Despite, I think the concept of a measure is much easier than that of a distribution. Therefore, I was wondering: In what sense is the concept of a Dirac distribution equivalent to the Dirac measure? Are you (in principle) able to prove all the properties of the distribution if you are using the concept of a measure? Or is the only thing that the Dirac delta measure is good for to say:
$$\int_{\mathbb{R}} f(x)\delta(x-x_0) d\mu(x):= \int_{\mathbb{R}} f(x)d\delta_{x_0}=f(x_0)?$$
Or differently: Would this definition be an appropriate definition? Or do we have to refer to the theory of distributions to prove all the properties that the Dirac-delta has?