Extending The Notion Of Riemann Integral To Lebesgue Integral On 2 R And Applications In Time Series Analysis.

This research work is intended for Senior undergraduate course in analysis ,’The 3rd and 4th \nyear B.ed and B.sc mathematics options’ and first year student mastering in mathematics. The \nproject covers topics in calculus ,real analysis, measure theory and applications in time series. \nThe beginning chapters lay the setting to Riemann integration in contrast with other earlier \nexisting theories such us mid-ordinate rule and Trapezium method. Riemann defines partition of \nindependent ordinate and take variation of the dependent ordinate then proceed to take the \nminimum and maximum sum of all the partitions possible and the integral is taken if the two \nRiemann sum are equal. Some examples of integration are also provided. The theory of Riemann \nstieltjes is an extension of Riemann theory that covers ;vector- valued functions and discontinuous \nfunctions such unit step functions and signum functions. It’s bridge the gap of continuity and \ndiscontinuity by use of convergence of series and also extend the real line to n R spaces. The \nfinal and most notable extension is the lebesgue integration. The construction of the lebesgue \nmeasure is done using countable base, whose members are open interval then the idea of \nmeasurable functions is extensively discussed ,before it’s use in definition of measurable integral \nis important ,the we proceed to define monotone convergence theorems and lebesgue dominated \nconvergence theorems. Finally the comparison of the two integration theories ‘Riemann and \nlebesgue’ is done by citing a number of similarity and loopholes in evaluation of integral in areas \nsuch as ;Bounded and Un bounded functions ,Complex and P L -spaces and recovery of derivative \nfunctions. Finally application of the Fourier Series integrals in Time-Series Analysis is done by \nby smoothing time plot by regression and other methods which allow finding of auto correlation , \nwavelet and spectrum analysis.