Editing Distribution (mathematics) (section)

===Radon measures===

The inclusion map <math>\operatorname{In} : C_c^\infty(U) \to C_c^0(U)</math> is a continuous injection whose image is dense in its codomain, so the [[transpose]] <math>{}^{t}\operatorname{In} : (C_c^0(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection.

Note that the continuous dual space <math>(C_c^0(U))'_b</math> can be identified as the space of [[Radon measure]]s, where there is a one-to-one correspondence between the continuous linear functionals <math>T \in (C_c^0(U))'_b</math> and integral with respect to a Radon measure; that is,

* if <math>T \in (C_c^0(U))'_b</math> then there exists a Radon measure <math>\mu</math> on {{mvar|U}} such that for all <math display=inline>f \in C_c^0(U), T(f) = \int_U f \, d\mu,</math> and
* if <math>\mu</math> is a Radon measure on {{mvar|U}} then the linear functional on <math>C_c^0(U)</math> defined by sending <math display=inline>f \in C_c^0(U)</math> to <math display=inline>\int_U f \, d\mu</math> is continuous.

Through the injection <math>{}^{t}\operatorname{In} : (C_c^0(U))'_b \to \mathcal{D}'(U),</math> every Radon measure becomes a distribution on {{mvar|U}}. If <math>f</math> is a [[locally integrable]] function on {{mvar|U}} then the distribution <math display=inline>\phi \mapsto \int_U f(x) \phi(x) \, dx</math> is a Radon measure; so Radon measures form a large and important space of distributions.

The following is the theorem of the structure of distributions of [[Radon measure]]s, which shows that every Radon measure can be written as a sum of derivatives of locally <math>L^\infty</math> functions on {{mvar|U}}:

{{math theorem|name='''Theorem.'''{{sfn|Trèves|2006|pp=262–264}}|math_statement=
Suppose <math>T \in \mathcal{D}'(U)</math> is a Radon measure, where <math>U \subseteq \R^n,</math> let <math>V \subseteq U</math> be a neighborhood of the support of <math>T,</math> and let <math>I = \{p \in \N^n : |p| \leq n\}.</math> There exists a family <math>f=(f_p)_{p\in I}</math> of locally <math>L^\infty</math> functions on {{mvar|U}} such that <math>\operatorname{supp} f_p \subseteq V</math> for every <math>p\in I,</math> and
<math display=block>T = \sum_{p\in I} \partial^p f_p.</math>
Furthermore, <math>T</math> is also equal to a finite sum of derivatives of continuous functions on <math>U,</math> where each derivative has order <math>\leq 2 n.</math>
}}

====Positive Radon measures====

A linear function <math>T</math> on a space of functions is called '''{{em|positive}}''' if whenever a function <math>f</math> that belongs to the domain of <math>T</math> is non-negative (that is, <math>f</math> is real-valued and <math>f \geq 0</math>) then <math>T(f) \geq 0.</math> One may show that every positive linear functional on <math>C_c^0(U)</math> is necessarily continuous (that is, necessarily a Radon measure).{{sfn|Trèves|2006|p=218}} 
[[Lebesgue measure]] is an example of a positive Radon measure.

====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.

====Test functions as distributions====

The test functions are themselves locally integrable, and so define distributions. The space of test functions <math>C_c^\infty(U)</math> is sequentially [[dense (topology)|dense]] in <math>\mathcal{D}'(U)</math> with respect to the strong topology on <math>\mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=300-304}} This means that for any <math>T \in \mathcal{D}'(U),</math> there is a sequence of test functions, <math>(\phi_i)_{i=1}^\infty,</math> that converges to <math>T \in \mathcal{D}'(U)</math> (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
<math display=block>\langle \phi_i, \psi \rangle \to \langle T, \psi \rangle \qquad \text{ for all } \psi \in \mathcal{D}(U).</math>