1 Introduction In the study of the derivation properties of interval functions there are certain arguments that reappear in many settings. It is our pupose here to present a unified approach to some of these techniques. In order to focus our attention we will present this material with a specific view to examining an interesting and important study of Rogers and Taylor [16] characterizing those interval functions which are in a sense absolutely continuous with respect to the s-dimensional Hausdorff measure. We follow Rogers [15] in confining our attention to the one-dimensional case although many of the arguments can be used in higher dimensions where richer geometry and larger choice of differentiation bases might obscure the natural simplicity of the ideas . The problem is to determine the nature of continuous, nondecreasing functions / on the interval [0,1] whose Lebesgue-Stieltjes measures Xj are absolutely continuous with respect to the s-dimensional Hausdorff measure on [0,1]. This problem leads naturally to an investigation of derivates of the form n*tf ^ r /(^) ~ f(y) m D{f,x)= hmsup (1) y,z^x,yxz [Z y ) s which Besicovich [1] has called "Lipschitz numbers". The discussion will be intimately related to the notion of a generalized Riemann integral as defined by Henstock [7] and Kurzweil [11]. Recall that the classical Riemann integral defined as a limit of Riemann sums rb n f(x)dx = \im^2 f(^i)(xi - xt-_i) (2) has been shown to allow greater flexibility than appears at a first study. By altering the definition of the limit in equation (2) one can obtain integrals that express exactly the improper Riemann integral, the Lebesgue integral, the Denjoy-Perron integral, the Denjoy-Khintchine integral, the approximate Perron and the approximate symmetric Perron integrals. The derivation theory for these integrals is particularly immediate for the same concept (what we call a "full covering relation" in Section 2) expresses both the integral and derivative, revealing a single unifying connection between these two classical ideas. In connection with the problem of Rogers and Taylor one might expect a similar approach to relate Hausdorff measures, Lipschitz numbers and ab- / J a 1
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