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
Canonical name	EveryLocallyIntegrableFunctionIsADistribution
Date of creation	2013-03-22 13:44:25
Last modified on	2013-03-22 13:44:25
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	8
Author	matte (1858)
Entry type	Theorem
Classification	msc 46-00
Classification	msc 46F05