TITLE: Was there ever proposed a theory where the value of Dirac Delta at zero had meaning on itself?
QUESTION [5 upvotes]: Was there ever proposed a theory where $\delta(0)$ has a meaningful value or used in a formal way outside integrals?
Particularly, following Fourier transforms, we can formally obtain
$$\pi\delta(0)=\int_0^\infty dx=\int_{0^+}^\infty \frac1{x^2}dx$$
Were there attempts to treat these integrals as some quantities?

REPLY [2 votes]: This works fine in Robinson's framework by choosing for example the Cauchy distribution (in the probability sense, not the Schwartz sense) with an infinitesimal value of the parameter.  The "infinitely tall, infinitely narrow" delta-function this obtained returns the value (of a test function at the point when integrated against it) up to an infinitesimal.  Robinson in his 1966 book proves the existence of delta functions which return the value of the test functions "on the nose" but these are a bit harder to define.