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[parent] every locally integrable function is a distribution (Theorem)

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)

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$.



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$T_f$ is a distribution of zeroth order (Proof) by Koro
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Cross-references: equivalence relation, injection, almost everywhere, parent, distribution, order, mapping, locally integrable function, open set
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This is version 5 of every locally integrable function is a distribution, born on 2003-07-09, modified 2006-01-30.
Object id is 4433, canonical name is EveryLocallyIntegrableFunctionIsADistribution.
Accessed 2564 times total.

Classification:
AMS MSC: 46F05 (Functional analysis :: Distributions, generalized functions, distribution spaces :: Topological linear spaces of test functions, distributions and ultradistributions)
 46-00 (Functional analysis :: General reference works )
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