every locally integrable function is a distribution

Suppose $U$ is an open set in $\\mathbb{R}^{n}$ and $f$ is a locallyintegrable 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 (seethis 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 withthe usual equivalence relation for an $L^{p}$ -space. For this reason,one usually writes $f$ for the distribution $T_{f}$ .