Editing Distribution (mathematics) (section)

====Locally integrable functions as distributions====

One particularly important class of Radon measures are those that are induced locally integrable functions. The function <math>f : U \to \R</math> is called '''{{em|[[locally integrable]]}}''' if it is [[Lebesgue integration|Lebesgue integrable]] over every compact subset {{mvar|K}} of {{mvar|U}}. This is a large class of functions that includes all continuous functions and all [[Lp space]] <math>L^p</math> functions. The topology on <math>\mathcal{D}(U)</math> is defined in such a fashion that any locally integrable function <math>f</math> yields a continuous linear functional on <math>\mathcal{D}(U)</math> – that is, an element of <math>\mathcal{D}'(U)</math> – denoted here by <math>T_f,</math> whose value on the test function <math>\phi</math> is given by the Lebesgue integral:
<math display=block>\langle T_f, \phi \rangle = \int_U f \phi\,dx.</math>

Conventionally, one [[Abuse of notation|abuses notation]] by identifying <math>T_f</math> with <math>f,</math> provided no confusion can arise, and thus the pairing between <math>T_f</math> and <math>\phi</math> is often written
<math display=block>\langle f, \phi \rangle = \langle T_f, \phi \rangle.</math>

If <math>f</math> and <math>g</math> are two locally integrable functions, then the associated distributions <math>T_f</math> and <math>T_g</math> are equal to the same element of <math>\mathcal{D}'(U)</math> if and only if <math>f</math> and <math>g</math> are equal [[almost everywhere]] (see, for instance, {{harvtxt|Hörmander|1983|loc=Theorem 1.2.5}}). Similarly, every [[Radon measure]] <math>\mu</math> on <math>U</math> defines an element of <math>\mathcal{D}'(U)</math> whose value on the test function <math>\phi</math> is <math display=inline>\int\phi \,d\mu.</math> As above, it is conventional to abuse notation and write the pairing between a Radon measure <math>\mu</math> and a test function <math>\phi</math> as <math>\langle \mu, \phi \rangle.</math> Conversely, as shown in a theorem by Schwartz (similar to the [[Riesz representation theorem]]), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.