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)