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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

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

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====
<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>