# every locally integrable function is a distribution

Suppose $U$ is an open set in $\mathbb{R}^{n}$ and $f$ is a locally integrable function on $U$, i.e., $f\in L^{1}_{\scriptsize{\mbox{loc}}}(U)$. Then the mapping

 $\displaystyle T_{f}:\mathcal{D}(U)$ $\displaystyle\to$ $\displaystyle\mathbb{C}$ $\displaystyle u$ $\displaystyle\mapsto$ $\displaystyle\int_{U}f(x)u(x)dx$

is a zeroth order distribution. (See parent entry for notation $\mathcal{D}(U)$.)

(proof) (http://planetmath.org/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 this page (http://planetmath.org/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 $T_{f}$.

Title every locally integrable function is a distribution EveryLocallyIntegrableFunctionIsADistribution 2013-03-22 13:44:25 2013-03-22 13:44:25 matte (1858) matte (1858) 8 matte (1858) Theorem msc 46-00 msc 46F05