An introduction to fractal Lebesgue integral

Abstract This manuscript explores various characteristics of generalized fractal measures. We expand the concept of fractal integrals in relation to step functions and examine their numerous properties. Notably, since all step functions are classified as simple functions, we apply the aforementioned generalized measure to introduce Lebesgue-type integrals, referred to as FL -integrals. Additionally, we demonstrate that all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>F</m:mi> <m:mi>α</m:mi> </m:msup> </m:math> {F^{\alpha}} -integrable functions are FL -integrals. Lastly, we address the bounded convergence theorem within the context of fractals.