A-integral

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)