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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}),
\end{displaymath} (1)

where $m$ is the Lebesgue Measure, and
I=\lim_{n\to\infty} \int_a^b [f(x)]_n\,dx
\end{displaymath} (2)

exists, where
f(x) & if $\vert f(x)\vert\leq n$\cr
0 & if $\vert f(x)\vert>n$.\cr}
\end{displaymath} (3)


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

© 1996-9 Eric W. Weisstein