# 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., $f\in {L}_{\text{loc}}^{1}(U)$. Then the mapping

${T}_{f}:\mathcal{D}(U)$ | $\to $ | $\u2102$ | ||

$u$ | $\mapsto $ | ${\int}_{U}}f(x)u(x)\mathit{d}x$ |

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 ${L}_{\text{loc}}^{1}$ 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 |