STUDIA MATHEMATICA 160 (1) (2004) The L r Henstock–Kurzweil integral by Paul M. Musial and Yoram Sagher (Chicago, IL) Abstract. We present a method of integration along the lines of the Henstock– Kurzweil integral. All L r -derivatives are integrable in this method. 1. Introduction. During the early part of the twentieth century, A. Denjoy and O. Perron developed equivalent integrals which extend the Lebesgue integral and which integrate all derivatives. Around 1960, R. Henstock and J. Kurzweil developed an integral which is equivalent to the integrals of Denjoy and Perron and which therefore integrates all derivatives. The construction of the Henstock–Kurzweil (HK) integral is quite similar to that of the Riemann integral and so is much easier than those of the Denjoy and Perron integrals. For a complete treatment of the Henstock–Kurzweil integral we refer the reader to [1], [4], or [5]. Instead of the classical derivative one may consider the approximate derivative, and with it integrals that will integrate such derivatives. Indeed, there exist both Perron-type and HK-type integrals that integrate approx- imate derivatives (see [4]). Another notion of derivative, the L r -derivative, useful in Harmonic Anal- ysis, was developed by A. P. Calder´ on and A. Zygmund in [2]. L. Gordon, in [3], developed the P r -integral (Perron r-integral) which integrates L r - derivatives. In this paper we define a Henstock–Kurzweil type integral which integrates all functions that are integrated by the P r -integral. We are considering functions defined on a finite closed interval, [a, b]. Also the parameter r, throughout the paper, satisfies 1 ≤ r< ∞. We use the phrase “nearly every” (abbreviated to n.e.) for “all but countably many.” Finally, the symbol , without a modifier, stands for the Lebesgue integral. 2. The P r -integral. To make our presentation reasonably self-contain- ed we give an outline of the P r -integral as developed by L. Gordon. For the full details of the proofs, see [3]. 2000 Mathematics Subject Classification : Primary 26A39, 26A42; Secondary 26A24, 26A46. [53]