2010-02-03

Measure, Lebesgue Integration, The Dirac Delta, Lebesgue-Stieltjes Integration and Cramer’s Representation

Dr. Roy Howard, Curtin University, Perth, Australia

Part 1: Singularity type functions, such as the Dirac delta, are usually defined assuming that they operate on well defined functions - usually called ‘good functions’. The results that hold for singular- ity functions, and for the good function case, do not hold, in general, for the usual case where non- good functions are to be considered and this has led to errors in published material. It will be shown how to define singularity functions for the general non-good function case. The following will also be clarified: Is the integral of the Dirac delta one or is it zero? Is the derivative of the unit step func- tion equal to the Dirac delta?

 

Part 2: Lebesgue-Stieltjes integration is not generally covered in Engineering programmes but is use- ful in many contexts. For example, it is used to define expectation for the case of random variables with both discrete and continuous outcomes and is fundamental to stochastic calculus and, accord- ingly, to many areas of engineering, physics, financial modelling etc. Further, it underpins harmonic analysis when Fourier analysis fails as is the case for stationary random phenomena on the infinite interval. For this case Cramer’s representation is useful. With the goal of understanding Cramer’s representation, an introduction to Lebesgue-Stieltjes integration will be given.

 

This presentation will be tutorial in nature.

Category: CE Seminar