Sub-probability measure

A sub-probability measure , also called a sub-probability distribution , is a set function in stochastics that represents a generalization of the probability measures . In contrast to probability measures, with sub-probability measures the superset is always assigned a number less than or equal to 1 and not exactly 1.

definition

A sub-probability measure is a set function

${\displaystyle \mu :{\mathcal {A}}\to [0,1]}$

on a measurement space , i.e. a basic set and a σ-algebra over this basic set with the following properties: ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$

• σ-additivity : For every countable sequence of pairwise disjoint setsfromtrue ${\displaystyle A_{1},A_{2},A_{3},\dots }$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\mu (A_{i})}$

• It is . ${\displaystyle \mu (X)\leq 1}$

Elementary properties

The finite signed dimensions over a common measurement space form a real vector space . In this space, the sub-probability measures contain the set of probability measures as a convex subset ; conversely, the sub-probability measures themselves form a convex subset of the finite measures and thus inherit many of their properties. As an example:

• It is ${\displaystyle \mu (\emptyset )=0}$
• Monotony : A sub-probability measure is a monotonous mapping from to , that is to say applies to ${\displaystyle ({\mathcal {A}},\subset )}$ ${\displaystyle ([0,1],\leq )}$ ${\displaystyle A,B\in {\mathcal {A}}}$

${\displaystyle B\subset A\implies \mu (B)\leq \mu (A)}$.

• σ-subadditivity : For any sequenceof sets inis true. ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)\leq \sum _{n=1}^{\infty }\mu (A_{n})}$
• σ-continuity from below : Isa monotonic against increasing set sequence in, so, so is. ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle A_{n}\uparrow A}$ ${\displaystyle \lim _{n\to \infty }\mu (A_{n})=\mu (A)}$
• σ-continuity from above : Ifa monotonic against decreasing set sequence in, so, then is. ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle A_{n}\downarrow A}$ ${\displaystyle \lim _{n\to \infty }\mu (A_{n})=\mu (A)}$

Properties on different floor spaces

The properties of sub-probability measures depending on the structure of the basic spaces ( topological space , metric space , Polish space , etc.) essentially correspond to the properties of finite measures on these spaces and are explained in detail in the article there.

One of the few differences between sub-probability measures and finite measures is that sequences or sets of sub-probability measures are always bounded. A sequence of dimensions is called limited if is. With the here is total variation norm referred. However, this is always true for sub-probability measures, since by definition ${\displaystyle \textstyle \sup _{n\in \mathbb {N} }\{\|\mu _{n}\|_{\operatorname {TV} }\}<\infty }$ ${\displaystyle \|\mu _{n}\|_{\operatorname {TV} }}$

${\displaystyle \mu (X)=\|\mu _{n}\|_{\operatorname {TV} }\leq 1}$

is. This leads, for example, to alternative formulations in Prokhorov's theorem , since then the restriction can be dispensed with. It then reads:

• If a separable metric space and a set of sub-probability measures on Borel's σ-algebra is tight , then the set is relatively compact in terms of weak convergence . ${\displaystyle (X,d)}$
• If a Polish space, then a set of sub-probability measures is relatively sequentially compact with respect to weak convergence if and only if the set is tight . ${\displaystyle X}$

Furthermore, there are special formulations of the Portmanteau theorem for sub-probability measures.

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 .