Suppose $U$ is an open set in $\sR^n$ and $f$ is a locally integrable function on $U$ , i.e., $f\in L^1_{\scriptsize{\mbox{loc}}}(U)$ . Then the mapping \begin{eqnarray*} T_f: \cD(U) &\to& \sC \\ u &\mapsto& \int_U f(x) u(x) dx \end{eqnarray*}is a zeroth order distribution. (See parent entry for notation $\cD(U)$ .)

(proof)

If $f$ and $g$ are both locally integrable functions on an open set $U$ , and $T_f=T_g$ , then it follows (see this page), that $f=g$ almost everywhere. Thus, the mapping $f\mapsto T_f$ is a linear injection when $L^1_{\scriptsize{\mbox{loc}}}$ is equipped with the usual equivalence relation for an $L^p$ -space. For this reason, one usually writes $f$ for the distribution $T_f$ .