Mathematical Notes, vol. 78, no. 2, 2005, pp. 228–233. Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 251–258. Original Russian Text Copyright c 2005 by P. Muldowney, V. A. Skvortsov. Improper Riemann Integral and Henstock Integral in R n P. Muldowney and V. A. Skvortsov Received May 12, 2004 Abstract—The Henstock integral in R n and its relation to the n-dimensional improper Rie- mann integral are studied. A Hake-type theorem for the Henstock integral in R n is proved. Key words: improper Rieman integral, Henstock integral, absolute integrability, Hake’s theo- rem, tagged partition, Riemann sum. In this note, we consider a Hake-type theorem for the Henstock integral in R n and use it to examine the relationship between this integral and the improper n-dimensional Riemann inte- gral. The following version of Hake’s theorem is known for the Henstock integral in R (see [1, Theorems 16.5 and 16.7]). Theorem A. A function f is Henstock integrable on R if and only if f is Henstock integrable on any compact interval [a, b] ⊂ R and there exists a finite limit lim a→−∞ lim b→∞  b a f = A. A particular case of this theorem is the following well-known fact. Theorem B. If f is integrable on R in the improper Riemann sense, then f is Henstock inte- grable on R , and the values of the integrals coincide. Theorem A shows also that the improper Henstock integral on R is equivalent to the Henstock integral on R (in sharp contrast to the Lebesgue integral on R). The general idea of defining the improper integral as a limit of integrals over increasing families {A α } of sets can be realized in the multidimensional case in several different ways depending on what family {A α } is chosen to generalize the compact intervals of the one-dimensional construction. It would be most natural to replace one-dimensional intervals by n-dimensional ones; that is, to define the improper Riemann integral in R n as lim a 1 ,...,a n →−∞ lim b 1 ,...,b n →∞  b 1 a 1  b 2 a 2 ···  b n a n f. (1) Unfortunately, with this definition of the improper integral in R n , Theorem B is no longer true in the n-dimensional case with n> 1 (see Example 1 below). Another (and most common) way to define the improper Riemann integral in R n (see [2] or [3]) predicates the use of a family {A α } of domains so wide that the resulting integral turns out to be absolutely convergent and, therefore, not a generalization of the one-dimensional improper integral. With such a definition, the corresponding version of Theorem B will of course be true, but will be a theorem about absolute integrals and will not, therefore, be a generalization of Theorem B, 228 0001-4346/2005/7812-0228 c 2005 Springer Science+Business Media, Inc.