From TheBestLinks.com
In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i.e. is a set with measure zero. If used for properties of the real numbers, the Lebesgue measure is assumed unless otherwise stated.
Occasionaly, instead of saying that a property holds almost everywhere, one also says that the property holds for almost all elements.
The term almost all in addition has several other meanings however.
Here is a list of theorems that involve the term "almost everywhere":
- If f : R -> R is a Lebesgue integrable function and f(x) ≥ 0 almost everywhere, then ∫ f(x) dx ≥ 0.
- If f : [a, b] -> R is a monotonic function, then f is differentiable almost everywhere.
- If f : R → R is Lebesgue measurable and ∫ab|f(x)|dx<∞ for every real numbers a<b then there exists a null set E (depending on f) such that, if x is not in E, the Lebesgue mean 1/(2e)∫x-ex+ef(t)dt converges to f(x) as e decreases to zero. In other words, the Lebesgue mean of f converges to f almost everywhere. The set E is called the Lebesgue set of f.
- If f(x,y) is Borel measurable on R2 then for almost every x, the function y→f(x,y) is Borel measurable.
Related links
Top visited
0 of
0 links
[no links posted yet]
>> place link >>
Discussion
Last posted
0 of
0 messages
[no messages posted yet]
>> post message >>
Watch
You can
add this article to your own "watchlist" and receive e-mail notification about all changes in this page.
