Measure theory and theory of the integral developed by Lebesgue at the beginning of the
last century found numerous applications in other branches of pure and applied mathematic for example in the theory of (partial) di erential equations, functional analysis and fractal
geometry; it is used to give mathematical foundation to probability theory and statistics, and
on the real line it gives a natural extension of the Riemann integral which allows for better
understanding of the fundamental relations between di erentiation and integration. This course
provides the essential foundations of this important aspect of mathematical analysis.