"Mutlak integrallenebilir fonksiyon" sayfasının sürümleri arasındaki fark
(Yeni sayfa: "==Definition and properties=== Consider a measure space <math>(X, \mathcal{A}, \mu)</math>. A measurable function <math>f:X \to [-\infty, \...") |
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| (Bir kullanıcı tarafından yapılan bir ara revizyon gösterilmiyor) | |||
| 1. satır: | 1. satır: | ||
| − | + | <math>\newcommand{\abs}[1]{\left|#1\right|}</math> | |
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| − | + | ===Tanım ve özellikler=== | |
| − | + | <math>(X, \mathcal{A}, \mu)</math> [[Ölçü uzayı|ölçü uzayı]] verilsin. Bir <math>f:X \to [-\infty, \infty]</math> [[Ölçülebilir fonksiyon|ölçülebilir fonksiyon]]una eğer | |
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| − | + | <math>\int \abs{f}\, d\mu < \infty\, .</math> | |
| − | <math> | + | |
| + | gerçeklerse mutlak integrallenebilirdir denir. | ||
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| + | Mutlak integrallenebilir bir fonksiyon genelde ''toplanabilir fonksiyon'' olarak da adlandırılır. | ||
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| + | '''Uyarı:''' | ||
| + | <math>|f|</math> mutlak değer fonksiyonunun ölçülebilirliği <math>f</math> fonksiyonunun ölçülebilirliğini garanti etmez. Bir çok yazar yalnızca <math>|f|</math> mutlak değer fonksiyonunun ölçülebilirliğini gerekli görürken büyük çoğunluk <math>f</math> fonksiyonunun kendisinin ölçülebilir olduğunu kabul eder. | ||
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| + | [[Jensen eşitsizliği|Jensen's eşitsizliği]]nin özel bir hali olan aşağıdaki eşitsizlik herhangi bir mutlak integrallenebilir fonksiyon için geçerlidir. | ||
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| + | <math> | ||
\abs{\int f\, d\mu}\leq \int \abs{f}\, d\mu | \abs{\int f\, d\mu}\leq \int \abs{f}\, d\mu | ||
| − | + | </math> | |
| + | |||
(the assumption of absolute integrability is however not fundamental: the inequality makes sense and holds as soon | (the assumption of absolute integrability is however not fundamental: the inequality makes sense and holds as soon | ||
as we can define | as we can define | ||
| − | <math> | + | |
| + | <math> | ||
\int f\, d\mu\, , | \int f\, d\mu\, , | ||
| − | + | </math> | |
| + | |||
that is, as soon as the integral of the positive part of <math>|f|</math> or that of the negative part of <math>|f|</math> are finite). | that is, as soon as the integral of the positive part of <math>|f|</math> or that of the negative part of <math>|f|</math> are finite). | ||
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| − | <math>L^1 (X, \mu)</math> | + | Mutlak integrallenebilir fonksiyonlar uzayı <math>L^1 (X, \mu)</math> ile gösterilen bir lineer uzaydır ve bu uzay üzerinde |
| − | <math> | + | |
| + | <math> | ||
\|f\|_1 := \int \abs{f}\, d\mu < \infty | \|f\|_1 := \int \abs{f}\, d\mu < \infty | ||
| − | + | </math> | |
| − | + | ||
| − | a | + | şeklinde bir yarınorm tanımlıdır. It is customary to identify elements of <math>L^1 (X, \mu)</math> whose values coincide except for |
| − | structure. The | + | a <math>\mu</math>-null set: after this identification the norme <math>\|\cdot\|_1</math> endowes <math>L^1 (X, \mu)</math> with a [[Banach space]] |
| + | structure. The <math>L^1</math> space is then just one case of a more general class of Banach spaces called | ||
[[Lp spaces]]. | [[Lp spaces]]. | ||
| − | === | + | ===Genelleştirmeler=== |
The notion of absolutely integrable function can be generalized to mappings taking values in normed vector spaces: | The notion of absolutely integrable function can be generalized to mappings taking values in normed vector spaces: | ||
in that case $\abs{\cdot}$ is substituted by the corresponding norm. This is straightforward for finite-dimensional | in that case $\abs{\cdot}$ is substituted by the corresponding norm. This is straightforward for finite-dimensional | ||
vector spaces and all the properties mentioned above holds in this case as well; for the case of infinite-dimensional spaces some care is needed, see [[Bochner integral]]. | vector spaces and all the properties mentioned above holds in this case as well; for the case of infinite-dimensional spaces some care is needed, see [[Bochner integral]]. | ||
| − | ===Lebesgue | + | ===Lebesgue ölçüsü=== |
The primary examples of absolutely integrable functions are given when <math>X</math> is an interval of the real axis (or | The primary examples of absolutely integrable functions are given when <math>X</math> is an interval of the real axis (or | ||
a domain of <math>\mathbb{R}^n</math>), <math>\mu</math> the [[Lebesgue measure]] and <math>\mathcal{A}</math> the corresponding [[Algebra of sets|<math>\sigma</math>-algebra]] of Lebesgue measurable functions. | a domain of <math>\mathbb{R}^n</math>), <math>\mu</math> the [[Lebesgue measure]] and <math>\mathcal{A}</math> the corresponding [[Algebra of sets|<math>\sigma</math>-algebra]] of Lebesgue measurable functions. | ||
| − | === | + | ===Genelleştirilmiş integral=== |
Consider an interval <math>[a,b[</math> (resp. <math>]a,b]</math>, <math>]a,b[</math>) where <math>b</math> might also be <math>\infty</math>. Some authors use the term | Consider an interval <math>[a,b[</math> (resp. <math>]a,b]</math>, <math>]a,b[</math>) where <math>b</math> might also be <math>\infty</math>. Some authors use the term | ||
absolutely integrable functions for functions <math>f</math> which are Riemann-integrable on all intervals <math>[a, \beta]</math> with <math>\beta<b</math> (resp. <math>]\alpha, b]</math>, <math>]\alpha, \beta[</math>) and for which | absolutely integrable functions for functions <math>f</math> which are Riemann-integrable on all intervals <math>[a, \beta]</math> with <math>\beta<b</math> (resp. <math>]\alpha, b]</math>, <math>]\alpha, \beta[</math>) and for which | ||
| − | <math> | + | |
| + | <math> | ||
\lim_{\beta\uparrow b} \int_a^\beta \abs{f (x)}\, dx < \infty\, | \lim_{\beta\uparrow b} \int_a^\beta \abs{f (x)}\, dx < \infty\, | ||
| − | + | </math> | |
| + | |||
(and analogous conditions for the other cases). | (and analogous conditions for the other cases). | ||
This implies the existence (and finiteness) of | This implies the existence (and finiteness) of | ||
| − | <math> | + | |
| + | <math> | ||
\lim_{\beta\uparrow b} \int_a^\beta f (x)\, dx\, | \lim_{\beta\uparrow b} \int_a^\beta f (x)\, dx\, | ||
| − | + | </math> | |
| + | |||
(and analogous limits for the other cases), which is often called [[Improper integral]]. | (and analogous limits for the other cases), which is often called [[Improper integral]]. | ||
The converse is not true, namely the existence of the improper integral does not guarantee the absolute integrability, regardless whether we are dealing with Riemann-integrable or Lebesgue-integrable functions. An example is | The converse is not true, namely the existence of the improper integral does not guarantee the absolute integrability, regardless whether we are dealing with Riemann-integrable or Lebesgue-integrable functions. An example is | ||
| − | <math> | + | |
| + | <math> | ||
\int_0^\infty \frac{\sin x}{x}\, dx\, . | \int_0^\infty \frac{\sin x}{x}\, dx\, . | ||
| − | + | </math> | |
| + | |||
| + | |||
| + | ===Kaynaklar=== | ||
| + | [1] C.D. Aliprantz, O. Burleinshaw, "Principles of real analysis" , North-Holland | ||
| + | |||
| + | [2] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , '''1–2''' , MIR (1982) (Translated from Russian) | ||
| + | |||
| + | [3] L.D. Kudryavtsev, "Mathematical analysis" , '''1''' , Moscow (1973) (In Russian) | ||
| + | |||
| + | [4] S.M. Nikol'skii, "A course of mathematical analysis" , '''1''' , MIR (1977) | ||
| + | |||
| + | [5] H.L. Royden, "Real analysis" , Macmillan (1968) | ||
| + | |||
| + | [6] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) | ||
| + | |||
| + | [7| W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 | ||
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| + | [8] L. Schwartz, "Cours d'analyse" , '''1''' , Hermann (1967) | ||
| + | |||
| + | [9] A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) | ||
| + | |||
| + | [10] A.C. Zaanen, "Integration" , North-Holland (1967) | ||
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14:15, 28 Kasım 2014 itibarı ile sayfanın şu anki hâli
$ \newcommand{\abs}[1]{\left|#1\right|} $
Konu başlıkları
Tanım ve özellikler
$ (X, \mathcal{A}, \mu) $ ölçü uzayı verilsin. Bir $ f:X \to [-\infty, \infty] $ ölçülebilir fonksiyonuna eğer
$ \int \abs{f}\, d\mu < \infty\, . $
gerçeklerse mutlak integrallenebilirdir denir.
Mutlak integrallenebilir bir fonksiyon genelde toplanabilir fonksiyon olarak da adlandırılır.
Uyarı: $ |f| $ mutlak değer fonksiyonunun ölçülebilirliği $ f $ fonksiyonunun ölçülebilirliğini garanti etmez. Bir çok yazar yalnızca $ |f| $ mutlak değer fonksiyonunun ölçülebilirliğini gerekli görürken büyük çoğunluk $ f $ fonksiyonunun kendisinin ölçülebilir olduğunu kabul eder.
Jensen's eşitsizliğinin özel bir hali olan aşağıdaki eşitsizlik herhangi bir mutlak integrallenebilir fonksiyon için geçerlidir.
$ \abs{\int f\, d\mu}\leq \int \abs{f}\, d\mu $
(the assumption of absolute integrability is however not fundamental: the inequality makes sense and holds as soon as we can define
$ \int f\, d\mu\, , $
that is, as soon as the integral of the positive part of $ |f| $ or that of the negative part of $ |f| $ are finite).
Mutlak integrallenebilir fonksiyonlar uzayı $ L^1 (X, \mu) $ ile gösterilen bir lineer uzaydır ve bu uzay üzerinde
$ \|f\|_1 := \int \abs{f}\, d\mu < \infty $
şeklinde bir yarınorm tanımlıdır. It is customary to identify elements of $ L^1 (X, \mu) $ whose values coincide except for a $ \mu $-null set: after this identification the norme $ \|\cdot\|_1 $ endowes $ L^1 (X, \mu) $ with a Banach space structure. The $ L^1 $ space is then just one case of a more general class of Banach spaces called Lp spaces.
Genelleştirmeler
The notion of absolutely integrable function can be generalized to mappings taking values in normed vector spaces: in that case $\abs{\cdot}$ is substituted by the corresponding norm. This is straightforward for finite-dimensional vector spaces and all the properties mentioned above holds in this case as well; for the case of infinite-dimensional spaces some care is needed, see Bochner integral.
Lebesgue ölçüsü
The primary examples of absolutely integrable functions are given when $ X $ is an interval of the real axis (or a domain of $ \mathbb{R}^n $), $ \mu $ the Lebesgue measure and $ \mathcal{A} $ the corresponding $ \sigma $-algebra of Lebesgue measurable functions.
Genelleştirilmiş integral
Consider an interval $ [a,b[ $ (resp. $ ]a,b] $, $ ]a,b[ $) where $ b $ might also be $ \infty $. Some authors use the term absolutely integrable functions for functions $ f $ which are Riemann-integrable on all intervals $ [a, \beta] $ with $ \beta<b $ (resp. $ ]\alpha, b] $, $ ]\alpha, \beta[ $) and for which
$ \lim_{\beta\uparrow b} \int_a^\beta \abs{f (x)}\, dx < \infty\, $
(and analogous conditions for the other cases). This implies the existence (and finiteness) of
$ \lim_{\beta\uparrow b} \int_a^\beta f (x)\, dx\, $
(and analogous limits for the other cases), which is often called Improper integral.
The converse is not true, namely the existence of the improper integral does not guarantee the absolute integrability, regardless whether we are dealing with Riemann-integrable or Lebesgue-integrable functions. An example is
$ \int_0^\infty \frac{\sin x}{x}\, dx\, . $
Kaynaklar
[1] C.D. Aliprantz, O. Burleinshaw, "Principles of real analysis" , North-Holland
[2] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[3] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian)
[4] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977)
[5] H.L. Royden, "Real analysis" , Macmillan (1968)
[6] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)
[7| W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98
[8] L. Schwartz, "Cours d'analyse" , 1 , Hermann (1967)
[9] A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965)
[10] A.C. Zaanen, "Integration" , North-Holland (1967)
