Final σ algebra

The final-σ-algebra , also called image-σ-algebra , is a special set system , more precisely a σ-algebra , in measure theory . The final σ-algebra formed for a given family of functions is the largest system of sets on the common target set of these functions, with respect to which they are all measurable . Thus, the concept of the final σ-algebra forms the counterpart to the concept of the initial-σ-algebra , which represents the smallest σ-algebra on the definition set, with respect to which all functions of the given family of functions can be measured. An analogous concept can be found in the topology ; Here the initial topology and the final topology are the coarsest or finest topology on the definition set or target set, with respect to which all functions of the given function family are continuous .

definition

For an arbitrary index set, measurement spaces are given as well as images for an arbitrary set . Then it is called σ-algebra ${\ displaystyle (X_ {i}, {\ mathcal {A}} _ {i})}$ ${\ displaystyle f_ {i} \ colon X_ {i} \ to X}$ ${\ displaystyle X}$

{\ displaystyle {\ mathcal {F}} (f_ {i}, \, i \ in I): = \ bigcap _ {i \ in I} \ {A \ subset X \, | \, f_ {i} ^ {-1} (A) \ in {\ mathcal {A}} _ {i} \}}

the final σ-algebra of the images on . ${\ displaystyle f_ {i}}$ ${\ displaystyle X}$

properties

Is another measuring space given and a function , it is precisely then - -measurable if the compositions all - are -measurable. ${\ displaystyle (Y, {\ mathcal {A}} ^ {*})}$ ${\ displaystyle g \ colon X \ to Y}$ ${\ displaystyle g}$ ${\ displaystyle {\ mathcal {F}} (f_ {i}, \, i \ in I)}$ ${\ displaystyle {\ mathcal {A}} ^ {*}}$ ${\ displaystyle g \ circ f_ {i}}$ ${\ displaystyle {\ mathcal {A}} _ {i}}$ ${\ displaystyle {\ mathcal {A}} ^ {*}}$

literature

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 .

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. 32 , doi : 10.1007 / 978-3-642-21026-6 .

[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. 32 , doi : 10.1007 / 978-3-642-21026-6 .