Full measure

A complete measure and a complete measure space are terms from measure theory , a branch of mathematics that deals with the generalization of volume concepts . A measure space is complete if it contains all subsets of its null sets. The dimension belonging to the dimension space is then called complete.

definition

Is a measure space , this is called complete if for any amount with all subsets in lying. If the measure space is complete, the measure is also called complete. If the smallest complete measure space that contains the measure space is called the completion of . ${\displaystyle (\Omega ,{\mathcal {A}}^{*},\mu ^{*})}$ ${\displaystyle N\in {\mathcal {A}}^{*}}$ ${\displaystyle \mu ^{*}(N)=0}$ ${\displaystyle A\subseteq N}$ ${\displaystyle {\mathcal {A}}^{*}}$ ${\displaystyle \mu ^{*}}$ ${\displaystyle M_{1}=(\Omega ,{\mathcal {A}}^{*},\mu ^{*})}$ ${\displaystyle M_{2}=(\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle M_{1}}$ ${\displaystyle M_{2}}$

Completion of dimensional spaces

If a measure space and the system of all subsets of -zero sets , then the measure space can be completed as follows: A second σ-algebra is defined as ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle {\mathcal {N}}}$ ${\displaystyle \mu }$

${\displaystyle {\tilde {\mathcal {A}}}:=\{A\cup N\,|\,A\in {\mathcal {A}},N\in {\mathcal {N}}\}}$

and a measure

${\displaystyle {\tilde {\mu }}(A\cup N):=\mu (A)}$.

Then the measure space is complete and even the smallest complete measure space that it contains. ${\displaystyle (\Omega ,{\tilde {\mathcal {A}}},{\tilde {\mu }})}$ ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$

Alternatively, you can also consider the external dimension generated by . If one restricts this to the σ-algebra of the measurable sets, then is a complete measure space. ${\displaystyle \mu }$ ${\displaystyle \mu ^{*}}$ ${\displaystyle {\mathcal {A}}_{\mu ^{*}}}$ ${\displaystyle \mu ^{*}}$ ${\displaystyle (\Omega ,{\mathcal {A}}_{\mu ^{*}},\mu ^{*}|_{{\mathcal {A}}_{\mu ^{*}}})}$

Examples

If an external measure is given and if the σ-algebra of the measurable sets and the corresponding measure are given, then the measure space is complete. This already follows from the definition of measurability, since if is with , then follows from the properties of the external measure and therefore . ${\displaystyle \nu }$ ${\displaystyle {\mathcal {A}}_{\nu }}$ ${\displaystyle \nu }$ ${\displaystyle \mu :=\nu |_{{\mathcal {A}}_{\nu }}}$ ${\displaystyle (\Omega ,{\mathcal {A}}_{\nu },\mu )}$ ${\displaystyle \nu }$ ${\displaystyle B\subset A\in {\mathcal {A}}_{\nu }}$ ${\displaystyle \nu (A)=0}$ ${\displaystyle \nu (B)=0}$ ${\displaystyle B\in {\mathcal {A}}_{\nu }}$

A well-known example of a completion is the completion of the Lebesgue-Borel measure to the Lebesgue measure . This completion also explains why the amount of Lebesgue measurable amounts is greater than that of Borel measurable amounts.

literature

• Otto Forster : Analysis. Volume 3: Measure and integration theory, integral theorems in R ⁿ and applications , 8th improved edition. Springer Spectrum, Wiesbaden, 2017, ISBN 978-3-658-16745-5 .
• Jürgen Elstrodt: Measure and integration theory. 6th edition. Springer, Berlin / Heidelberg / New York 2009, ISBN 978-3-540-89727-9 .
• Achim Klenke: Probability Theory. 2nd Edition. Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-76317-8 .