Event system

An event system , also called event algebra , event space or event field , is a set system in stochastics that contains all sets to which a probability is to be assigned. These sets are then also called events . The restriction to a set system that is smaller than the power set of the result space is based on negative statements such as Vitali's theorem that not all elements of the power set can be meaningfully assigned a measure and thus a probability.

definition

A result space is given that contains all possible results of a modeled random experiment. Then is a σ-algebra on the ground set an event system , an algebra of events , event space or event field . ${\ displaystyle \ Omega}$ ${\ displaystyle \ Sigma}$ ${\ displaystyle \ Omega}$

Sometimes the couple is also referred to as an event space, which corresponds to a measurement space in the sense of measurement theory. ${\ displaystyle (\ Omega, \ Sigma)}$

interpretation

The following requirements are fundamental when modeling a random experiment:

You want to be able to assign a probability of 1 to the fact that something happens. So a probability must be assignable to the superset and it must therefore be in the event set. ${\ displaystyle \ Omega}$
If one can assign a probability to an event, one also wants to be able to assign a probability to the fact that this event does not occur. So it must also be in the event set. ${\ displaystyle A}$ ${\ displaystyle A ^ {\ mathrm {c}} = \ Omega \ setminus A}$
If a countable number of events occur, the event that at least one of these events occurs should also be in the event set. This is precisely the union of the countable many . ${\ displaystyle (A_ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle A_ {n}}$

An event set does not have to be too large in order to avoid non-measurable sets, but it must be stable with respect to these operations in order to enable meaningful modeling. The set system that fulfills these requirements is a σ-algebra , which is used canonically to model sets of events.

Examples

Let's look at the result set , it has the three results ${\ displaystyle \ Omega = \ {1,2,3 \}}$ ${\ displaystyle \ omega _ {1} = 1, \ omega _ {2} = 2, \ omega _ {3} = 3}$

One of the possible event systems would be

{\ displaystyle \ Sigma _ {1}: = \ {\ Omega, \ emptyset, \ {1 \}, \ {2,3 \} \}}

.

It should be noted that the corresponding event does not necessarily have to be included in the event system for every result . ${\ displaystyle \ omega _ {i}}$ ${\ displaystyle \ {\ omega _ {i} \}}$

Canonical event systems

Finite or countably infinite result sets

On finite or countably infinite result sets one always chooses the power set as the event system , since it is easy to handle and in this case does not lead to any paradoxes. For example, one equips the result set of the natural numbers with the event system . ${\ displaystyle \ mathbb {N}}$ ${\ displaystyle {\ mathcal {P}} (\ mathbb {N})}$

Real result set

If the result set is the set of real numbers or an uncountable subset of, for example , it is always equipped with Borel's σ-algebra or the correspondingly restricted trace σ-algebra . These event systems are smaller than the power sets, but contain all sets that can be naively constructed. Borel's σ-algebra can also be defined for any topological space . ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle [0,1]}$

Result sets as products

If the result sets are products of several sets, then one always chooses the product σ-algebra as the event system.

classification

The following hierarchy applies:

Results are elements of the result set and the events ${\ displaystyle \ omega}$
Events are subsets of the result set and elements of the event system. They contain results as elements.
Event systems are subsets of the power set.

In particular, a distinction must be made between the result and the event . ${\ displaystyle \ omega}$ ${\ displaystyle \ {\ omega \}}$

literature

Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 .
Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .

Individual evidence

↑ Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 195 , doi : 10.1007 / 978-3-642-21026-6 .
↑ David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 59 , doi : 10.1007 / b137972 .
^ Georgii: Stochastics. 2009, p. 10.

[Schmidt195-1] Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 195 , doi : 10.1007 / 978-3-642-21026-6 .

[2] David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 59 , doi : 10.1007 / b137972 .

[3] Georgii: Stochastics. 2009, p. 10.