outer measure
Definition [1, 2, 3]
Let be a set, and let be the
power set of . An outer measure
on is a function
satisfying the properties
-
1.
.
-
2.
If are subsets in , then .
-
3.
If is a countable
collection
of subsets of , then
Here, we can make two remarks. First, from (1) and (2), it follows
that is a positive function on . 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.
References
-
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.
Title | outer measure |
---|---|
Canonical name | OuterMeasure |
Date of creation | 2013-03-22 13:45:20 |
Last modified on | 2013-03-22 13:45:20 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 6 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 60A10 |
Classification | msc 28A10 |
Related topic | CaratheodorysExtensionTheorem |
Related topic | CaratheodorysLemma |
Related topic | ProofOfCaratheodorysExtensionTheorem |