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One of the generalizations of the Lebesgue integral, given by E. Titchmarsh [[#References|[1]]] for the integration of functions conjugate to summable ones. A measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a0100102.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a0100104.png" />-integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a0100105.png" /> if
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One of the generalizations of the Lebesgue integral, given by E. Titchmarsh [[#References|[1]]] for the integration of functions conjugate to summable ones. A measurable function $f(x)$ is called $A$-integrable over $[a,b]$ if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a0100106.png" /></td> </tr></table>
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$$m\{x\colon|f(x)|>n\}=O\left(\frac1n\right)$$
  
 
and if
 
and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a0100107.png" /></td> </tr></table>
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$$I=\lim_{n\to\infty}\int\limits_a^b[f(x)]_ndx$$
  
 
exists, where
 
exists, where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a0100108.png" /></td> </tr></table>
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$$[f(x)]_n=\begin{cases}f(x)&\text{if }|f(x)|\leq n,\\0&\text{if }|f(x)|>n.\end{cases}$$
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a0100109.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a01001011.png" />-integral. It is denoted by
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The number $I$ is called the $A$-integral. It is denoted by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010010/a01001012.png" /></td> </tr></table>
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$$(A)\quad\int\limits_a^bf(x)dx.$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.G. Titchmarsh,  "On conjugate functions"  ''Proc. London Math. Soc.'' , '''29'''  (1928)  pp. 49–80</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Vinogradova,  "Generalized integrals and Fourier series"  ''Itogi Nauk. Mat. Anal. 1970''  (1971)  pp. 65–107  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.G. Titchmarsh,  "On conjugate functions"  ''Proc. London Math. Soc.'' , '''29'''  (1928)  pp. 49–80</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Vinogradova,  "Generalized integrals and Fourier series"  ''Itogi Nauk. Mat. Anal. 1970''  (1971)  pp. 65–107  (In Russian)</TD></TR></table>

Revision as of 12:01, 27 October 2014

One of the generalizations of the Lebesgue integral, given by E. Titchmarsh [1] for the integration of functions conjugate to summable ones. A measurable function $f(x)$ is called $A$-integrable over $[a,b]$ if

$$m\{x\colon|f(x)|>n\}=O\left(\frac1n\right)$$

and if

$$I=\lim_{n\to\infty}\int\limits_a^b[f(x)]_ndx$$

exists, where

$$[f(x)]_n=\begin{cases}f(x)&\text{if }|f(x)|\leq n,\\0&\text{if }|f(x)|>n.\end{cases}$$

The number $I$ is called the $A$-integral. It is denoted by

$$(A)\quad\int\limits_a^bf(x)dx.$$

References

[1] E.G. Titchmarsh, "On conjugate functions" Proc. London Math. Soc. , 29 (1928) pp. 49–80
[2] I.A. Vinogradova, "Generalized integrals and Fourier series" Itogi Nauk. Mat. Anal. 1970 (1971) pp. 65–107 (In Russian)
How to Cite This Entry:
A-integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A-integral&oldid=18741
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article