## Extending the Notion of Riemann Integral to Lebesgue Integral on R2 and Applications in Time Series Analysis.

##### Abstract

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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