Continuous function with compact carrier

A continuous function with a compact support is a special continuous function that only takes the value 0 outside of a compact unit . Such functions play an important role in functional analysis , as well as in stochastics and measure theory , where they are used as a separating family for sets of measures and the definition of convergence concepts.

definition

A topological space and a standardized space as well as a mapping are given ${\displaystyle (X,\tau )}$ ${\displaystyle (S,\|\cdot \|_{S})}$

${\displaystyle f\colon X\to S}$.

The mapping is called a continuous function with compact support if the support of the function, i.e. the set ${\displaystyle f}$

${\displaystyle \operatorname {supp} (f):={\overline {\{x\in X\mid f(x)\neq 0\}}}}$

is a compact set and the map is continuous . It is therefore true that the archetypes of open sets (with regard to the topology generated by) are open again under , i.e. are contained in. If there is a metric space, this means that for all sequences that converge against , the image sequence converges against . ${\displaystyle \|\cdot \|_{S}}$ ${\displaystyle f}$ ${\displaystyle \tau }$ ${\displaystyle X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x_{0}}$ ${\displaystyle (f(x_{n}))_{n\in \mathbb {N} }}$ ${\displaystyle f(x_{0})}$

The set of all continuous functions with compact support is usually denoted by or . If it is clear which rooms are involved, you do not specify them; accordingly, the designations or are often found ${\displaystyle C_{c}(X,S)}$ ${\displaystyle C_{c}^{0}(X,S)}$ ${\displaystyle S=\mathbb {K} }$ ${\displaystyle C_{c}(X)}$ ${\displaystyle C_{c}^{0}(X)}$

structure

If one defines the addition and the scalar multiplication in point-wise, that is ${\displaystyle C_{c}(X;S)}$

${\displaystyle (f+g)(x):=f(x)+g(x){\text{ sowie }}(\lambda f)(x):=\lambda f(x){\text{ für alle }}x\in X}$,

such is a vector space. ${\displaystyle C_{c}(X;S)}$

Furthermore, every continuous function with a compact support is also a constrained function .

For example is a metric space, then there exists due to the continuity of each item on a so ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle \epsilon _{x}}$

${\displaystyle f(B_{\epsilon _{x}}(x))\subset B_{1}(f(x))}$

If the carrier is now covered by the open sets , then due to the compactness there is a finite index set , so that the carrier is covered. Thus applies ${\displaystyle f}$ ${\displaystyle (B_{\epsilon _{x}}(x))_{x\in \operatorname {supp} (f)}}$ ${\displaystyle I}$ ${\displaystyle \left(B_{\epsilon _{x_{i}}}(x_{i})\right)_{i\in I}}$

${\displaystyle f(X)=f(\operatorname {supp} (f))\subset f\left(\bigcup _{i\in I}B_{\epsilon _{x_{i}}}(x_{i})\right)\subset \bigcup _{i\in I}B_{1}(f(x_{i}))}$.

So is limited. is thus a subspace of , the space of limited mappings . For topological spaces one can generalize this argumentation with the help of a covering of the support with sets of the form . ${\displaystyle f}$ ${\displaystyle C_{c}(X,S)}$ ${\displaystyle B(X,S)}$ ${\displaystyle f^{-1}(B_{1}(f(x)))}$

Because of the limitation, the definition of the supremum norm is through ${\displaystyle C_{c}(X;S)}$

${\displaystyle \|f\|_{\operatorname {sup} }:=\sup _{x\in X}\|f(x)\|_{S}}$

makes sense and makes a standardized space. ${\displaystyle C_{c}(X;S)}$

Superordinate structures

${\displaystyle C_{c}(X)}$is a subspace of , the space of continuous, infinitely vanishing functions and bounded continuous functions , so the implications hold ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{b}(X)}$

${\displaystyle B(X)\supset C_{b}(X)\supset C_{0}(X)\supset C_{c}(X)}$.

In addition, for a locally finite measure (or Borel measure ) on a Hausdorff space, every continuous function with a compact support can also be integrated, there ${\displaystyle \mu }$ ${\displaystyle X}$

${\displaystyle \int _{X}f\mathrm {d} \mu =\int _{X}f\chi _{K}\mathrm {d} \mu \leq \mu (K)\|f\|_{\infty }<\infty }$

as a result of local finiteness. So in this case . ${\displaystyle \mu (K)<\infty }$ ${\displaystyle C_{c}(X)\subset {\mathcal {L}}^{1}(X,{\mathcal {A}},\mu )}$

Subordinate structures

The test functions are an important subspace of the continuous functions with a compact carrier .

Important statements

According to Riesz-Markow's theorem, any positive linear form can be used in a locally compact Hausdorff space ${\displaystyle X}$

${\displaystyle I\colon C_{c}(X)\to \mathbb {K} }$

represent as

${\displaystyle I(f)=\int f\mathrm {d} \mu }$,

where is a uniquely determined radon level . A linear form is called positive if it always follows. ${\displaystyle \mu }$ ${\displaystyle f\geq 0}$ ${\displaystyle I(f)\geq 0}$

literature

• Hans Wilhelm Alt : Linear Functional Analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , doi : 10.1007 / 978-3-642-22261-0 .
• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .