Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT
| MMA0005 | Real Functions |
| Teaching Staff in Charge |
Assoc.Prof. ANISIU Valeriu, Ph.D., anisiu math.ubbcluj.roProf. BULBOACA Teodor, Ph.D., bulboaca math.ubbcluj.ro |
| Aims |
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Learning the fundamental facts in general topology, measure and integartion
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| Content |
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Topological spaces, neighborhoods, closure, interior, boundary
Bases and subbases, generated topology, subspace, product spaces, convergence, continuity. Separation exioms (T1,T2), metric spaces, completeness. Compactness. Etension of continuous functions. Baire functions. Connectedness. Applications to the extended real numbers, extreme limits. Algebras and sigma-algebras. Measures. Outer measures, the Lebesgue outer measure in R^m. Construction of the Lebesgue measure. Regularity. Measurable functions, simple functions, a.e.-convergence. The Lebesgur integral. Applications. Lebesgue vs. Riemann integrals. |
| References |
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[1] V. Anisiu: Topologie si teoria masurii. Universitatea $Babes-Bolyai$, Cluj-Napoca, 1995.
[2] C. Crăciun : Lecţii de analiză matematică. Universitatea Bucureşti, 1982. [3] C. Crăciun : Exerciţii şi probleme de analiză matematică. Universitatea Bucureşti 1984. [4] C. George: Exercises in integration. Springer, New York, 1984 [5] J. Kelley: General topology. Van Nostrand, Princeton, 1950. [6] P. Kree: Integration et theorie de la mesure. Une approche geometrique. Ellipses, Paris, 1997 [7] W. Rudin: Real and complex analysis, McGraw Hill, New York, 1988 (exista traducere in limba romana) [8] G.B. Folland: Real Analysis. Modern Techniques and their applications. Wiley, 1999 |
| Assessment |
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Midterm test and Final Exam.
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| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |