Boll. di mat. pura ed appl. Vol. IV (2011) The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria Elettrica, Elettronica e delle Telecomunicazioni, di Tecnologie Chimiche, Automatica e Modelli Matematici (DIEETCAM) Università degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy E-mail addresses: donatella.bongiorno@unipa.it, giusi.corrao@unipa.it Abstract The aim of this paper is to introduce an Henstock-Kurzweil-Stieltjes type integration process for real functions on a fractal subset E of tha real line. Key words: s-set; s-derivatives; s-integral Contents 1 Introduction 1 2 Preliminaries 3 2.1 The s-integral ................................ 3 3 The s-Henstock-Kurzweil-Stieltjes integral 5 3.1 Linearity properties ............................ 7 3.2 Cauchy Criterion and Cauchy extension ................. 9 3.2.1 Saks-Henstock Lemma ....................... 11 4 Relationship between the s-integral and the s-Henstock-Kurzweil- Stieltjes integral 15 4.1 The Vitali-Carathéodory Theorem .................... 15 1 Introduction During last years, some mathematicians have been obliged to define a new concept of derivation (see [1] and [2]) and a new concept of integration (see, [6] and [9]) to solve some physical and engineering problems. In fact even if the geometry of fractal D. Bongiorno, G. Corrao (pp. ? – ??) 1