PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Medium Entry average rating: No information on entry rating
complete measure (Definition)

A measure space $ (X,\mathscr{S},\mu)$ is said to be complete if every subset of a set of measure 0 is measurable (and consequently, has measure 0); i.e. if for all $ E\in\mathscr{S}$ such that $ \mu(E)=0$ and for all $ S\subset E$ we have $ \mu(S)=0$.

If a measure space is not complete, there exists a completion of it, which is a complete measure space $ (X,\overline{\mathscr{S}},\overline{\mu})$ such that $ \mathscr{S}\subset\overline{\mathscr{S}}$ and $ \overline {\mu}_{\vert\mathscr{S}} = \mu$, where $ \overline{\mathscr{S}}$ is the smallest $ \sigma$-algebra containing both $ \mathscr{S}$ and all subsets of elements of zero measure of $ \mathscr{S}$.



"complete measure" is owned by Koro.
(view preamble)

View style:

Also defines:  completion, complete

Attachments:
completion of a measure space (Derivation) by Koro
Log in to rate this entry.
(view current ratings)

Cross-references: measurable, measure, subset, measure space
There are 31 references to this entry.

This is version 2 of complete measure, born on 2004-01-18, modified 2004-01-18.
Object id is 5520, canonical name is CompleteMeasure.
Accessed 5676 times total.

Classification:
AMS MSC28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)