Principal value of $1/x$ does not arise from either a locally integrable function or a Radon measure

Question

The distribution $\text{p.v.}1/x\in C_c^\infty(\mathbb{R})^*$ is defined by the formula: $$\left\langle f,\text{p.v.}\frac{1}{x}\right\rangle:=\lim_{\varepsilon \to 0}\int_{|x|>\varepsilon} \frac{f(x)}{x}\,dx.$$ How to show that this distribution does not arise from either a locally integrable function or a Radon measure? It seems that we need to choose appropriate test functions to get a contradiction (see this post).

Answer 1

If a distribution $T$ arises from a $L^1_{\rm loc}$ function or a Radon measure, then there is a constant $M$ such that $$|T\phi|\le M\sup_{[-1,1]}|\phi|\tag{1}$$ for all test functions supported on $[-1,1]$.

But (1) fails for p.v. $1/x$, as can be seen by considering a test function that is supported on $[-1,1]$, is equal to $1$ on $[\epsilon, 1/2]$, and is between $0$ and $1$ everywhere.