Strong differentiation of an indefinite integral

Finding the strong derivative of an indefinite integral

of a real-valued function that is summable in an open subset of -dimensional Euclidean space, considered as a function of the interval . If

is summable on (in particular, if , ), then the integral of is strongly differentiable almost-everywhere on . For any , , that is positive, non-decreasing and such that

as , there is a summable function on such that is also summable and such that the ratio is unbounded at each , as tends to , that is, cannot be strongly differentiated.

References

[1]	B. Jessen, J. Marcinkiewicz, A. Zygmund, "Note on the differentiability of multiple integrals" Fund. Math. , 25 (1935) pp. 217–234
[2]	S. Saks, "On the strong derivatives of functions of intervals" Fund. Math. , 25 (1935) pp. 235–252
[3]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[4]	A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)

References

[a1]	A. Zygmund, "On the differentiability of multiple integrals" Fund. Math. , 23 (1934) pp. 143–149