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: Very high Entry average rating: No information on entry rating
outer measure (Definition)

Definition [1,2,3] Let $ X$ be a set, and let $ \mathcal{P}(X)$ be the power set of $ X$. An outer measure on $ X$ is a function $ \mu^\ast:\mathcal{P}(X)\to [0,\infty]$ satisfying the properties

  1. $ \mu^\ast(\emptyset)=0$.
  2. If $ A\subset B$ are subsets in $ X$, then $ \mu^\ast(A)\le \mu^\ast(B)$.
  3. If $ \{A_i\}$ is a countable collection of subsets of $ X$, then
    $\displaystyle \mu^\ast(\bigcup_i A_i) \le \sum_i \mu^\ast (A_i).$

Here, we can make two remarks. First, from (1) and (2), it follows that $ \mu^\ast$ is a positive function on $ \mathcal{P}(X)$. Second, property (3) also holds for any finite collection of subsets since we can always append an infinite sequence of empty sets to such a collection.

Bibliography

1
A. Mukherjea, K. Pothoven, Real and Functional analysis, Plenum press, 1978.
2
A. Friedman, Foundations of Modern Analysis, Dover publications, 1982.
3
G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.



"outer measure" is owned by mathcam. [ owner history (1) ]
(view preamble)

View style:

Log in to rate this entry.
(view current ratings)

Cross-references: empty sets, sequence, infinite, finite, positive, collection, countable, subsets, properties, function, power set
There is 1 reference to this entry.

This is version 3 of outer measure, born on 2003-07-15, modified 2005-05-17.
Object id is 4456, canonical name is OuterMeasure2.
Accessed 3509 times total.

Classification:
AMS MSC28A10 (Measure and integration :: Classical measure theory :: Real- or complex-valued set functions)
 60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory)

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

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