Separating family

In stochastics and measure theory, a separating family is a set of measurable images that can be used to distinguish certain measures. Separating families occur, for example, in the definition of convergence concepts of measures or the definition of completeness in mathematical statistics.

definition

Given a set of Radon measures on and a lot of measurable figures . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle E}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle E\to \mathbb {R} }$

${\displaystyle {\mathcal {F}}}$is then called a separating family (or simply separating) for , if the following applies to all : ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \mu _{1},\mu _{2}\in {\mathcal {M}}}$

If   for everyone   , then is   . ${\displaystyle \int f\mathrm {d} \mu _{1}=\int f\mathrm {d} \mu _{2}}$ ${\displaystyle f\in {\mathcal {F}}\cap {\mathcal {L}}(\mu _{1})\cap {\mathcal {L}}(\mu _{2})}$ ${\displaystyle \mu _{1}=\mu _{2}}$

The dimensions can therefore be differentiated using the integrals over the function class.

Examples

Evidence that a set of functions is divisive is usually more time-consuming. For example:

• If a metric space , then the set of all Lipschitz continuous mappings from to with Lipschitz constant 1 (also referred to as ) separates the set of radon measures. ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle [0,1]}$ ${\displaystyle \operatorname {Lip} _{1}(E,[0,1])}$
• If, in addition to the above prerequisites , it is also locally compact , the amount is decisive for the amount of radon measures. Here denotes the set of all continuous functions with a compact carrier . ${\displaystyle E}$ ${\displaystyle C_{c}(E)\cap \operatorname {Lip} _{1}(E,[0,1])}$ ${\displaystyle C_{c}(E)}$

application

An example of the application of the separating families is the definition of convergence terms. Since the dimension is clearly defined by the separating family, the following convergence term is suitable for a set of radon dimensions and an associated separating family : ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle \lim _{n\to \infty }\mu _{n}=\mu \;:\iff \lim _{n\to \infty }\int f\mathrm {d} \mu _{n}=\int f\mathrm {d} \mu {\text{ für alle }}f\in {\mathcal {F}}}$

Examples for this are:

• The weak convergence of measures defines the convergence of finite measures on a metric space provided with Borel's σ-algebra . The set of bounded continuous functions is chosen as the separating family .
• The vague convergence of dimensions thus defines the convergence of radon dimensions on a locally compact Hausdorff space. The set of continuous functions on a compact carrier is chosen as a separating family .

Similar statements can also be found in the context of the Portmanteau theorem for characterizing the convergence of dimensions.

literature

• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .