# Difference between revisions of "A-integral"

From Encyclopedia of Mathematics

(TeX, to be added: {{MSC|26A42}}) |
|||

(2 intermediate revisions by 2 users not shown) | |||

Line 1: | Line 1: | ||

{{TEX|done}} | {{TEX|done}} | ||

− | 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 | + | 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 |

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

## Latest revision as of 11:01, 31 March 2017

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=34099

This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article