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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| $  A $-
integral]] proved to be more convenient for the integration of functions conjugate with summable functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>