Bologna Information System

  • Course Information
  • Course Title Code Semester Laboratory+Practice (Hour) Pool Type ECTS
    Learning Outcomes
    1-To determine the measurability of a set.
    2-To express the properties of measure function.
    3-To compute the integral of a measurable simple function.
    4-To explain the cases for which riemann integral is insufficient and the usefulness of lebesgue integral.
    5- to explain convergence in l_p and in measure, the relation between these convergences.
  • ActivityPercentage


    NumberTime (Hours)Total Workload (hours)
    Course Duration (Weeks x Course Hours)14456
    Classroom study (Pre-study, practice)14684
    Short-Term Exams (exam + preparation) 1021428
    Midterm exams (exam + preparation)3011616
    Laboratory 0000
    Final exam (exam + preparation) 5011818
    Other 0000
    Total Workload (hours)   230
    Total Workload (hours) / 30 (s)     7,67 ---- (8)
    ECTS Credit   8
  • Course Content
  • Week Topics Study Metarials
    1 Preliminaries, countable and uncountable sets, functions, sequences, convergence of sequences
    2 bounded sequences, sequence of sets and their limits, some set classes, sigma Ring and sigma algebras
    3 Borel algebras, measurable set, measure function and its properties
    4 outer measure and Lebesgue outer measure, Lebesgue measure
    5 measurable functions, generating measurable functions from measurable functions
    6 integral of simple functions, integral of positive functions
    7 Monotone Convergence Theorem, Fatou lemma, Beppo-Levi Theorem
    8 Lebesgue integral, Absolute integrability property of Lebesgue integral
    9 Tchebichev inequality, Charge function and its properties Lebesgue co
    10 nvergence theorem and its applications, Bounded Convergence Theorem and its applications
    11 The relation between Riemannian integral and Lebesgue integral
    12 L_p spaces and its properties
    13 Hölder and Minkowski inequalities, Riesz-Fischer Theorem
    14 L infinity spaces and its properties, uniform convergence, convergence in measure and Lp- convergence.
    Prerequisites -
    Language of Instruction Turkish
    Course Coordinator Assist. Prof. Dr. Gülsüm ULUSOY
    Instructors -
    Assistants -
    Resources Introduction to real analysis, M. Stoll, Addison Wesley, 2000.
    Supplementary Book [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.
    Document Lecture notes.
    Goals To analyze properties of measure theory, Lebesgue integral and L_p spaces in real numbers set.
    Content -
  • Program Learning Outcomes
  • Program Learning Outcomes Level of Contribution
    1 To have a grasp of theoretical and applied knowledge in main fields of mathematics 5
    2 To have the ability of abstract thinking 4
    3 To be able to use the gained mathematical knowledge in the process of identifying the problem, analyzing and determining the solution steps 5
    4 To be able to relate the gained mathematical acquisitions with different disciplines and apply in real life 2
    5 To have the qualification of studying independently in a problem or a project requiring mathematical knowledge 3
    6 To be able to work compatibly and effectively in national and international groups and take responsibility -
    7 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 5
    8 To be able to determine what sort of knowledge the problem met requires and guide the process of learning this knowledge 4
    9 To adopt the necessity of learning constantly by observing the improvement of scientific accumulation over time 3
    10 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 4
    11 To be able to produce projects and arrange activities with awareness of social responsibility -
    12 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 -
    13 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 -
    14 To have the awareness of acting compatible with social, scientific, cultural and ethical values -
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