σ-additivity

The σ-additivity, sometimes also called countable additivity , is in stochastics and measure theory a property of functions that are defined on systems of sets , whose arguments are sets. It is essential for the modern axiomatic structure of stochastics as well as the theory of measure and integration, but is also rejected by some mathematicians such as Bruno de Finetti .

definition

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

${\displaystyle f\colon {\mathcal {M}}\to \mathbb {R} \cup \{+\infty \}}$

ie σ-additive, if for each countable series of pairwise disjoint sets out , for the back , is ${\displaystyle A_{1},A_{2},A_{3},\dotsc }$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \bigcup _{i=1}^{\infty }A_{i}}$ ${\displaystyle {\mathcal {M}}}$

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

applies.

Remarks

It should be noted that the requirement is not that the system of sets is closed with respect to countable unions, but only that the above equation should apply if the countable union is again in the system of sets.

example

Every measure and every premeasure is σ-additive by definition.

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 .