Boks integral

One of the generalizations of the Lebesgue integral, proposed by A. Denjoy (1919) and studied in detail by T.J. Boks (1921). A real-valued function $f$ on a segment $[a, b]$ is periodically extended (with period length $b - a$) to the entire straight line. For an arbitrary subdivision $a = x _ {0} < x _ {1} < \dots < x _ {n} = b$ of $[a, b]$, for an arbitrary selection of points $\overline \xi \; = \{ \xi _ {i} \} _ {1} ^ {n}$, $\xi _ {i} \in [x _ {i-1 } , x _ {i} ]$, and an arbitrary $t$, the following sum is constructed:

$$I(t) = \sum _ { i=1 } ^ { n } f( \xi _ {i} +t) [x _ {i} -x _ {i-1} ].$$

If, for $\rho = \max _ {i} (x _ {i} - x _ {i-1 } ) \rightarrow 0$, $I(t)$ converges in measure to a definite limit $I$, the number $I$ is said to be the Boks integral ( $B$- integral) of $f$ over $[a, b]$. Thus, the Boks integral is an integral of Riemann type and is a generalization of the Riemann integral.

The Boks integral represents a considerable extension of the Lebesgue integral: Any summable function is $B$- integrable and these integrals coincide, but there exist non-summable $B$- integrable functions; in particular, if $g$ is the function conjugate with a summable function $f$, then it is $B$- integrable and the coefficients of the series conjugate with the Fourier series of $f$ are the coefficients of the respective Fourier series (in the sense of $B$- integration) of $g$( A.N. Kolmogorov). The theory of the Boks integral was not further developed, since the $A$- integral proved to be more convenient for the integration of functions conjugate with summable functions.

References

[1]	T.J. Boks, "Sur les rapports entre les méthodes de l'intégration de Riemann et de Lebesque" Rend. Circ. Mat. Palermo (2) , 45 (1921) pp. 211–264
[2]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)