A locally integrable function annihilating every compactly supported continuous test function is zero almost everywhere.Suggest a slogan
Let \(f\colon\RR^n\to\CC\) be locally integrable, that is, \(\int_K\abs{f}<\infty\) for every compact \(K\subseteq\RR^n\), and let \(C_c^0(\RR^n)\) be the space of continuous functions \(\RR^n\to\CC\) with compact support. If \[ \int_{\RR^n} f \phi = 0 \quad \text{for all } \phi\in C_c^0(\RR^n), \] then \(f = 0\) almost everywhere.