I'm trying to figure out how to show distributions agree with a given function on some domain.
For instance, let $f \in C(\mathbb{R^n}\setminus\{0\})$ such that $f(rx) = r^{-n}f(x)$ and $\int f d\sigma = 0$ (where $\sigma$ is the surface measure on the sphere). These conditions imply that $f$ is not locally integrable near the origin, but we can define the principal value distribution
$$(PV(f), \phi) = \lim_{\epsilon -> 0} \int_{|x| > \epsilon} f(x)\phi(x)dx$$
which agrees with $f$ on $\mathbb{R^n}\setminus\{0\}$. I'm not sure how I should show this fact - that the principal value distribution agrees with $f$. Any help to point me toward the right direction would be appreciated.
Take any compact $K$ not containing zero, then take any test function $\phi$ with support in $K$, then $$\langle pv(f),\phi\rangle = \lim_{\epsilon\to 0}\int_{\|x\|\ge \epsilon}f(x)\phi(x)dx.$$ If $$\epsilon< dist(K,0),$$then $$ \int_{\|x\|\ge \epsilon}f(x)\phi(x)dx = \int_Kf(x)\phi(x)dx,$$hence $$\langle pv(f),\phi\rangle = \int_{K}f(x)\phi(x)dx$$ and therefore you have the desired result.