Extending the Notion of Riemann Integral to Lebesgue Integral on R2 and Applications in Time Series Analysis.
This research work is intended for Senior undergraduate course in analysis ,’The 3rd and 4th year B.ed and B.sc mathematics options’ and first year student mastering in mathematics. The project covers topics in calculus ,real analysis, measure theory and applications in time series. The beginning chapters lay the setting to Riemann integration in contrast with other earlier existing theories such us mid-ordinate rule and Trapezium method. Riemann defines partition of independent ordinate and take variation of the dependent ordinate then proceed to take the minimum and maximum sum of all the partitions possible and the integral is taken if the two Riemann sum are equal. Some examples of integration are also provided. The theory of Riemann stieltjes is an extension of Riemann theory that covers ;vector- valued functions and discontinuous functions such unit step functions and signum functions. It’s bridge the gap of continuity and discontinuity by use of convergence of series and also extend the real line to n R spaces. The final and most notable extension is the lebesgue integration. The construction of the lebesgue measure is done using countable base, whose members are open interval then the idea of measurable functions is extensively discussed ,before it’s use in definition of measurable integral is important ,the we proceed to define monotone convergence theorems and lebesgue dominated convergence theorems. Finally the comparison of the two integration theories ‘Riemann and lebesgue’ is done by citing a number of similarity and loopholes in evaluation of integral in areas such as ;Bounded and Un bounded functions ,Complex and P L -spaces and recovery of derivative functions. Finally application of the Fourier Series integrals in Time-Series Analysis is done by by smoothing time plot by regression and other methods which allow finding of auto correlation , wavelet and spectrum analysis.
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