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Week
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Topics
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Study Metarials
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1
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Preliminaries, countable and uncountable sets, functions, sequences, convergence of sequences
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2
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bounded sequences, sequence of sets and their limits, some set classes, sigma Ring and sigma algebras
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3
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Borel algebras, measurable set, measure function and its properties
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4
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outer measure and Lebesgue outer measure, Lebesgue measure
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5
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measurable functions, generating measurable functions from measurable functions
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6
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integral of simple functions, integral of positive functions
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7
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Monotone Convergence Theorem, Fatou lemma, Beppo-Levi Theorem
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8
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Lebesgue integral, Absolute integrability property of Lebesgue integral
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9
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Tchebichev inequality, Charge function and its properties
Lebesgue co
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10
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nvergence theorem and its applications, Bounded Convergence Theorem and its applications
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11
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The relation between Riemannian integral and Lebesgue integral
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12
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L_p spaces and its properties
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13
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Hölder and Minkowski inequalities, Riesz-Fischer Theorem
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14
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L infinity spaces and its properties, uniform convergence, convergence in measure and Lp- convergence.
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Prerequisites
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-
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Language of Instruction
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Turkish
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Course Coordinator
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Assist. Prof. Dr. Gülsüm ULUSOY
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Instructors
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Assistants
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Resources
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Introduction to real analysis, M. Stoll, Addison Wesley, 2000.
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Supplementary Book
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[1] Reel Analiz, Mustafa Balcı, Ertem Matbaası, 2000.
[2] Measure, Integral and Probability, Capinski, M . ve Kopp, E. Springer,1999.
[3] Lebesgue Measure and Integration, P. K. Jain and V. P. Gupta, John Willey and Sons Inc., 1996.
[4] Real analysis with Real Applications, K. R. Davidson, A. P. Donsig, Prentice Hall, 2002.
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Document
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Lecture notes.
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Goals
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To analyze properties of measure theory, Lebesgue integral and L_p spaces in real numbers set.
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Content
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Program Learning Outcomes |
Level of Contribution |
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1
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To have a grasp of theoretical and applied knowledge in main fields of mathematics
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5
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2
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To have the ability of abstract thinking
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4
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3
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To be able to use the gained mathematical knowledge in the process of identifying the problem, analyzing and determining the solution steps
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5
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4
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To be able to relate the gained mathematical acquisitions with different disciplines and apply in real life
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2
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5
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To have the qualification of studying independently in a problem or a project requiring mathematical knowledge
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3
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6
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To be able to work compatibly and effectively in national and international groups and take responsibility
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-
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7
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To be able to consider the knowledge gained from different fields of mathematics with a critical approach and have the ability to improve the knowledge
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5
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8
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To be able to determine what sort of knowledge the problem met requires and guide the process of learning this knowledge
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4
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9
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To adopt the necessity of learning constantly by observing the improvement of scientific accumulation over time
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3
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10
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To be able to transfer thoughts on issues related to mathematics, proposals for solutions to the problems to the expert and non-expert shareholders written and verbally
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4
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11
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To be able to produce projects and arrange activities with awareness of social responsibility
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12
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To be able to follow publications in mathematics and exchange information with colleagues by mastering a foreign language at least European Language Portfolio B1 General Level
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-
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13
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To be able to make use of the necessary computer softwares (at least European Computer Driving Licence Advanced Level), information and communication technologies for mathematical problem solving, transfer of thoughts and results
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14
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To have the awareness of acting compatible with social, scientific, cultural and ethical values
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