GENERALIZED HAKE'S PROPERTY FOR A KURZWEIL HENSTOCK TYPE INTEGRAL
c a b c f If f is integrable on [a,b] in the improper Lebesgue sense, then f is Kurzweil Henstock integrable on [a,b] and the values of the integrals coincide. The general idea of computing the improper integral as a limit of integral over increasing families {Aa} of sets can be realized in the multidimensional case in several different ways depending on the type of integral and on what family {Aa} is chosen to generalize the compact intervals of the above onedimensional construction. It would be most natural to replace onedimensional intervals by n dimensional ones; that is, to define improper integral in R n as Unfortunately, with this definition of the improper integral in R n the statement is no longer true in n dimensional case with n>1 (see [6] Another way to define the improper integral in R n predicates the use of the family {A a of domains so wide (in particular arbitrary Jordan measurable sets) that the resulting integrals turns out to be absolutely convergent and, therefore, not a generalization of the one dimensional improper integral (see [4] With such a definition, the corresponding version of will of course be true, but will be a theorem about absolute integrals and will not, therefore, be a generalization of which is about non absolute integrals. Thus, in this case the concept of an improper integral does not lead to anything new as compared with Lebesgue integral. So the family of n dimensional compact intervals is too small and the family of all Jordan measurable sets is too large to obtain an integral for which a proper generalization of holds. So the problem is to find a suitable class of increasing families {Aa} for computing integral keeping the Hake property for multidimensional nonabsolute integral. These classes are different for Kurzweil Henstock type integrals defined by different derivation basis. This fact gives rise to various type of Hake theorem. Some version of it for certain Kurzweil Henstock type integrals in R n was studied in [2] [3] and [6] n n n n c a c a c a b b c c