The Hake's property for some integrals over multidimensional intervals (Q1898967)

The Hake's property for a one-dimensional Perron-Denjoy integral is the equivalence between the integrability of a function over \([a,b]\) and its integrability over \([a,c]\) for all \(c \in ]a,b[\) and the existence of the limit \(\lim_{c \to b -} \int^c_af\). For a real function \(f\) defined on a compact interval \(I\) of \(\mathbb{R}^n\), and for various types of multidimensional nonabsolute integrals, this note proves the equivalence between the integrability of \(f\) over \(I\) and the integrability of \(f\) over all intervals \(J \subset \text{int} I\) together with the existence of the appropriate limit \(\lim_{F \to \text{int} I} \int_Ff\), where \(F \subset \text{int} I\) is a finite union of compact intervals of \(\mathbb{R}^n\).