Two questions about distributions

Question

1. Is it correct to assume that: $$\int f(x)\,P.v.\left(\frac{1}{x-a}\right)\delta(x-a)\,dx\,=\,0$$ ? Where the delta function is taken as a measure and $f(x)$ is a test function (it might be good to know what conditions it has to satisfy), and P.v. denotes the principal value.

2. I know that squaring a delta function is undefined object. However, in what sense would it be possible to just say that the dagger operation (Hermitian conjugate) leaves this undefined object unaffected when it appears inside an integral (that contains compact support)?

Answer 1

The value of that expression isn't well-defined.

Taking $f(x)=x$ we have on one hand, $$ \int f(x) \, \operatorname{pv}\frac{1}{x} \, \delta(x) \, dx = \int x \, \operatorname{pv}\frac{1}{x} \, \delta(x) \, dx = \int 1 \, \delta(x) \, dx = 1 $$ since $x \, \operatorname{pv}\frac{1}{x} = 1$ (constant), and on the other hand, $$ \int f(x) \, \operatorname{pv}\frac{1}{x} \, \delta(x) \, dx = \int \operatorname{pv}\frac{1}{x} \, x\, \delta(x) \, dx = \int \operatorname{pv}\frac{1}{x} \, 0 \, dx = 0 $$ since $x \, \delta(x) = 0.$

Contradiction!