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 (211)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (38)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
almost everywhere (Definition)

Let $ (X, \mathfrak{B}, \mu)$ be a measure space. A condition holds almost everywhere on $ X$ if it holds “with probability $ 1$,” i.e. if it holds everywhere except for a subset of $ X$ with measure 0. For example, let $ f$ and $ g$ be nonnegative functions on $ X$. Suppose we want a sufficient condition on functions $ f(x)$ and $ g(x)$ such that the relation

$\displaystyle \int_{X} f d\mu(x) \le \int_{X} g d\mu(x)$ (1)

holds. Certainly $ f(x)\leq g(x)$ for all $ x\in X$ is a sufficient condition, but in fact it's enough to have $ f(x)\leq g(x)$ almost surely on $ X$. In fact, we can loosen the above non-negativity condition to only require that $ f$ and $ g$ are almost surely nonnegative as well.

If $ X = [0,1]$, then $ g$ might be less than $ f$ on the Cantor set, an uncountable set with measure 0, and still satisfy the condition. We say that $ f \le g$ almost everywhere (often abbreviated a.e.).

Note that this term is the equivalent of the term “almost surely” from probabilistic measure theory.



"almost everywhere" is owned by mathcam. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

Other names:  almost surely, a.s., a.e., almost all

Attachments:
continuous almost everywhere versus equal to a continuous function almost everywhere (Example) by Wkbj79
Log in to rate this entry.
(view current ratings)

Cross-references: uncountable set, Cantor set, relation, sufficient, functions, measure, subset, measure space
There are 78 references to this entry.

This is version 4 of almost everywhere, born on 2002-02-16, modified 2004-04-09.
Object id is 2002, canonical name is AlmostSurely.
Accessed 20892 times total.

Classification:
AMS MSC: 60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory)
Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

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