A-Integrable

A generalization of the Lebesgue Integral. A Measurable Function $f(x)$ is called $A$-integrable over the Closed Interval $[a,b]$ if

$$m\{x: \vert f(x)\vert>n\}={\mathcal O}(n^{-1}),$$ (1)

where $m$ is the Lebesgue Measure, and

$$I=\lim_{n\to\infty} \int_a^b [f(x)]_n\,dx$$ (2)

exists, where

$$[f(x)]_n=\cases{ f(x) & if $\vert f(x)\vert\leq n$\cr 0 & if $\vert f(x)\vert>n$.\cr}$$ (3)

References

Titmarsch, E. G. ``On Conjugate Functions.'' Proc. London Math. Soc. 29, 49-80, 1928.