σ-subadditivity

In measure theory, σ-subadditivity is a property of a set function , i.e. a function whose arguments are sets.

definition

A system of sets is given on the basic set , that is . An illustration ${\displaystyle {\mathcal {M}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {M}}\subset {\mathcal {P}}(X)}$

${\displaystyle f\colon {\mathcal {M}}\to \left[0,\infty \right]}$

is called σ-subadditive if for every sequence of sets from and each with that ${\displaystyle A_{1},A_{2},\dots }$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle A\in {\mathcal {M}}}$ ${\displaystyle A\subset \bigcup _{i=1}^{\infty }A_{i}}$

${\displaystyle f(A)\leq \sum _{i=1}^{\infty }f(A_{i})}$

is. Note that it is not necessary here to request. ${\displaystyle \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {M}}}$

Examples

Every external dimension is σ-subadditive by definition. For premeasures on rings (and thus also for measures on σ-algebras ) the σ-subadditivity results from the defining property of the σ-additivity .

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 .
• 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 .

Individual evidence

1. ↑ Achim Klenke: Probability Theory. Springer-Verlag, 2013, ISBN 978-3-642-36018-3 , p. 12 ( limited preview in the Google book search).