The modern notion of measure, developed in the late 19th century, is an extension of the notions of length, area or volume. A *measure* *m* is a law which assigns a number to certain subsets *A*
of a given space and is a natural generalization of the following
notions: 1) length of an interval, 2) area of a plane figure, 3) volume
of a solid, 4) amount of mass contained in a region, 5) probability that
an event from *A* occurs, etc.

It originated in the real analysis and is used now in many areas of mathematics like, for instance, geometry, probability theory, dynamical systems, functional analysis, etc.

Given a measure *m*, one can define the *integral* of suitable real valued functions with respect to *m*.
Riemann integral is applied to continuous functions or functions with
``few`` points of discontinuity. For measurable functions that can be
discontinuous ``almost everywhere'' Riemann integral does not make
sense. However it is possible to define more flexible and powerful *Lebesgue's integral* (integral with respect to *Lebesgue's measure*) which is one of the key notions of modern analysis.

The course will cover the following topics: Definition of a measurable space and -additive
measures, Construction of a measure form outer measure, Construction of
Lebesgue's measure, Lebesgue-Stieltjes measures, Examples of
non-measurable sets, Measurable Functions, Integral with respect to a
measure, Lusin's Theorem, Egoroff's Theorem, Fatou's Lemma, Monotone
Convergence Theorem, Dominated Convergence Theorem, Product Measures and
Fubini's Theorem. Selection of advanced topics such as Radon-Nikodym
theorem, covering theorems, differentiability of monotone functions
almost everywhere, descriptive definition of the Lebesgue integral,
description of Riemann integrable functions, *k*-dimensional measures in *n*-dimensional spaces, divergence theorem, Riesz representation theorem, etc.

- Teacher: Dr. Javed Hussain