STATH208

Stat H 208 : Real Analysis 3 credits

1. The real number system; axioms and completeness and its consequences; sets, compact sets; simple operation on them, cluster (limit) points; Bolzano-Weierstrass theorem.

• 1. Infinite sequences; Convergence; Theorems on limits; Monotone sequences; subsequences; Cauchy sequences; Cauchy’s general principle of convergence; Cauchy’s first and second theorems on limits; infinite series of real numbers; convergence, and absolute convergence; tests for convergence.

3. Continuity; continuous functions; uniform continuity; Intermediate value theorem; Mean value theorem; Taylor expansion (with remainder or in infinite series).

4. Metric and topological spaces; limit points; open and closed sets; interior and exterior points; boundary points; Continuous mapping and Cauchy sequences.

5. Riemann-Steiljes integrals via Riemann’s sums and Darboux’s sums. Necessary and sufficient conditions for integrability.

6. Measure and integrals on abstract sets on real lines; Cramer measurability: fundamental definitions; auxiliary lemma; fundamental theorems; Measurable functions; Lebesgue measure on a real line, plane; Distinction between probability measure and Lebesque measure and Lebesque integrals.

7. Examples of applications in Statistics.

Text:

1. Robert G. Bartle, Donald R. Sherbert. Introduction to real analysis, 4th edition.

References:

1. Royden, H.L., Real Analysis, Mcmillan,N.Y.

2. Schaum Series, Advanced Calculus & Real Analysis.

3. Sipschute, S., General Topol;ogy, McGraw-Hill, N.Y.

4. Halmos,P.R., Measure Theory, Van Nostrand, N. Y.

5. Billingeley, P., Probabability and Measure, Wiley, N.Y.

6. Kingman, J.F.G., Measure and Probability, CUP.

7. Pitt, H.R., Integration measure and Probability, Oliver and Boyed.

8. William, F.T.(2011): Introduction to Real Analysis, Peason Publication, USA.

9. V Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.

10. Simmons, G.F,, Introduction to Topology & Modern Analysis, McGraw-Hill,N.Y.

11. Berherion S.K., Introduction to Measure & Integration.