I am trying to show that the delta distribution cannot be written in terms of an integral. To do this, we assume otherwise and deduce a contradiction.
If it were so, we would have for all test functions , so in particular for the test function defined by if , and zero otherwise, and so we would have
The book I am reading then argues that if \delta were a locally integrable function, Lebesgue's general theorem of convergence would tell us that the integral converges to 0.
I cannot see how this can be deduced by the theorem. Can anyone help me out?