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
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] support of integrable function with respect to counting measure is countable (Result)

Let $ (X,\mathfrak{B},\mu)$ be a measure space with $ \mu$ the counting measure. If $ f$ is an integrable function, $ \displaystyle \int_X f\,d\mu<\infty$, then it has countable support.

Proof. WLOG, we assume that $ f$ is real valued and is nonnegative. Let $ S_0$ denote the preimage of the interval $ [1,\infty)$ and, for every positive integer $ n$, let $ S_n$ denote the preimage of the interval $ \left[\frac{1}{n+1},\frac{1}{n}\right)$. Since the integral of $ f$ is bounded, each $ S_n$ can be at most finite. Taking the union of all the $ S_n$, we get the support $ \displaystyle \operatorname{supp}f = \bigcup_{n=0}^\infty S_n$. Thus, $ \operatorname{supp}f$ is a union of countably many finite sets and hence is countable. $ \qedsymbol$



"support of integrable function with respect to counting measure is countable" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: uncountable sums of positive numbers


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: finite sets, union, finite, bounded, integral, integer, positive, interval, preimage, real, WLOG, function, counting measure, measure space
There are 2 references to this entry.

This is version 8 of support of integrable function with respect to counting measure is countable, born on 2005-02-01, modified 2007-08-13.
Object id is 6698, canonical name is SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable.
Accessed 1214 times total.

Classification:
AMS MSC: 28A12 (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 example | add (any)