Arago spot: Difference between revisions

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en>Srleffler
Rv. There is no "aperture" here. Diffraction around a circular obscuring object is considered.
en>Samatict
Edited revision 451506797 by La goutte de pluie (talk) to clean up grammar, external links, and phrasing; neutralized weighted word choices
 
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{{no footnotes|date=March 2013}}
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In [[measure theory]] (a branch of [[mathematical analysis]]), a property holds '''almost everywhere''' if the set of elements for which the property does not hold is a set of measure zero (Halmos 1974). In cases where the measure is not [[Complete measure|complete]], it is sufficient that the set is contained within a set of measure zero. When discussing sets of [[real number]]s, the [[Lebesgue measure]] is assumed unless otherwise stated.  
 
The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent [[French language]] phrase ''presque partout''.
 
A set with '''full measure''' is one whose complement is of measure zero. In [[probability theory]], the terms ''[[almost surely]]'', ''almost certain'' and ''almost always'' refer to sets with [[probability]] 1, which are exactly the sets of full measure in a probability space.  
 
Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for '''almost all''' elements (though the term [[almost all]] also has other meanings).
 
== Properties ==
 
* If ''f'' : '''R''' → '''R''' is a [[Lebesgue integral|Lebesgue integrable]] function and ''f''(''x'') ≥ 0 almost everywhere, then
 
::<math>\int_a^b f(x) \, dx \geq 0</math>
 
:for all real numbers ''a'' < ''b'' with equality [[iff]] <math>f(x)=0</math> almost everywhere.
 
* If ''f'' : [''a'', ''b''] → '''R''' is a [[monotonic function]], then ''f'' is [[derivative|differentiable]] almost everywhere.
* If ''f'' : '''R''' → '''R''' is Lebesgue measurable and
 
::<math>\int_a^b |f(x)| \, dx < \infty</math>
 
:for all real numbers ''a'' < ''b'', then there exists a set ''E'' (depending on ''f'') such that, if ''x'' is in ''E'', the Lebesgue mean
 
::<math>\frac{1}{2\epsilon} \int_{x-\epsilon}^{x+\epsilon} f(t)\,dt</math>
 
:converges to ''f''(''x'') as <math>\epsilon</math> decreases to zero. The set ''E'' is called the Lebesgue set of ''f''. Its complement can be proved to have measure zero. In other words, the Lebesgue mean of ''f'' converges to ''f'' almost everywhere.
 
* If ''f''(''x'',''y'') is [[Borel measurable]] on '''R'''<sup>2</sup> then for almost every ''x'', the function ''y''→''f''(''x'',''y'') is Borel measurable.
 
* A bounded [[function (mathematics)|function]] ''f'' : [''a'', ''b''] <tt>-></tt> '''R''' is [[Riemann integral|Riemann integrable]] if and only if it is [[continuous function|continuous]] almost everywhere.
 
== Definition using ultrafilters ==
 
Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of an [[ultrafilter]]. An ultrafilter on a set ''X'' is a maximal collection ''F'' of subsets of ''X'' such that:
# If ''U'' ∈ ''F'' and ''U'' ⊆ ''V'' then ''V'' ∈ ''F''
# The intersection of any two sets in ''F'' is in ''F''
# The empty set is not in ''F''
A property ''P'' of points in ''X'' holds almost everywhere, relative to an ultrafilter ''F'', if the set of points for which ''P'' holds is in ''F''.  
 
For example, one construction of the [[hyperreal number]] system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter.
 
The definition of ''almost everywhere'' in terms of ultrafilters is closely related to the definition in terms of measures, because each ultrafilter defines a finitely-additive measure taking only the values 0 and 1, where a set has measure 1 if and only if it is included in the ultrafilter.
 
==References==
* {{cite book
| last = Billingsley
| first = Patrick
| authorlink =
| year = 1995
| title = Probability and measure
| edition = 3rd
| publisher = John Wiley & Sons
| location = New York
| isbn = 0-471-00710-2
}}
* {{cite book
| last = Halmos
| first = Paul R.
| authorlink = Paul Halmos
| year = 1974
| title = Measure Theory
| publisher = Springer-Verlag
| location = New York
| isbn = 0-387-90088-8
}}
 
[[Category:Measure theory]]
[[Category:Mathematical terminology]]
 
[[de:Maßtheorie#fast überall]]
[[ja:ほとんど (数学)#ほとんど至るところで]]

Latest revision as of 14:09, 6 October 2014

Pushups helps to tone up the chest muscles to a great extent. encourages your body to create a scar tissue capsule around the. Why are there so many ways to approach the subject of cup size. Consumers can't know if the product they are taking is actually herbal. Always choose quality FDA-approved implants for better results.

Small, underdeveloped breasts may affect self-confidence. To some degree, the amount of time needed will depend on how and where the implant is being placed. You can buy them to show more tempting, bigger breast. Dicalcium Phosphate - Dicalcium Phosphate is manufactured from phophoric acid. Normally, ladies frequently rely upon ONLY ONE SOURCE for comprehensive breast augmentation -- and, especially in regard to Fenugreek breast enlargement, herein lay the most central problems that cause countless individuals to miss the mark of natural breast enhancement achievement.

used to be the case that cheap breast augmentation prices were. In addition to breast enlargement, it is used for other cosmetic surgery procedures, in particular the plumping up of facial features, cheeks and chin and is a popular alternative to other wrinkle treatments. You must not fall in a pitfall while undergoing the surgery at a cheaper rate. Fenugreek aids in hormonal production, and facilitates the development of the mammary glands which 'feed' on estrogens. To date there is no conclusive evidence of a particular type of implant being absolutely better that another, and the choice at the time of consultation and surgery is dependent on the Surgeons assessment.

Hold this position for a count of four, and then slowly return to starting position. Kim, who became an overnight celebrity in 2007 after her X-rated sex tape mysteriously surfaced online, insists her late dad, the attorney Robert Kardashian, would be pleased with her success. Nursing is a wonderful and necessary activity, but it can lead to sagging, drooping breasts. Estrogen-based breast augmentation products may cause a temporary enhancement as many women have discovered when using birth control pills. Using natural creams and supplements or pills is a very safe and natural elective to follow other than going under the knife.

Breast enlargement exercises may give you a bigger measurement but cup size and breast size will not be increased. Pushup or padded bras are one option, but there are other seemingly possible ways, as well. It can be taken by anyone regardless of age or gender, and does not contain any addictive components. We live in a culture where good looks are considered a must, especially for all those who want to move up the ladder. Sexual desires and personal interest enforce the women to adopt breast enlarging techniques.

The best is they offer a full 100% Lifetime Money Back, so that you can take your time to see and evaluate the results. You need to keep in your mind, during breast enlargement you have to take protein diets and strictly avoid caffeine because caffeine can be harmful to your body. Every woman, who has planned to go with the surgery, seeks out certain set of questions that are to be solved. Breast Massage is a conventional way to improve their breast size. Relaxing habits (such as taking a hot bath or reading a book) give your body the signal that it's time to rest.

For those who have any kind of concerns concerning where along with the best way to make use of breast increase, you can e-mail us with the web-site.

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