EveryLocallyIntegrableFunctionIsADistribution 2003-07-09 05:13:30 2006-01-30 06:58:09 Theorem distribution % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} \newcommand{\cD}[0]{\mathcal{D}} 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 $\cD(U)$.) \PMlinkname{(proof)}{T_fIsADistributionOfZerothOrder} If $f$ and $g$ are both locally integrable functions on an open set $U$, and $T_f=T_g$, then it follows (see \PMlinkname{this page}{TheoremForLocallyIntegrableFunctions}), 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 $T_f$.