Required in stage
First term
Second term
Both terms
Not organised
This year
Next year
Alternating years
External
Prerequisites
Taught by
Language of instruction
Duration
Identical coursesAims
After following this course:
(1) the student is able to outline Lebesgue's theory of integration in the general context of an arbitrary measure space,
(2) the student is able to state the measure theoretical foundations of probability theory, and is able to illustrate them at the level of examples,
(3) the student is able to identify the classical theorems of measure theory and he/she recognizes situations in analysis and probability theory where those results can be applied,
(4) the student is familiar with some classical techniques from measure theory and theoretical probability theory, and he/she is able to apply these techniques to relatively new situations,
(5) the student has further developed his/her sense of generality and abstraction,
(6) the student has further sharpend his/her abiltity to construct proofs,
(7) the student has further developed his/her (self-)critical sense of accuracy and clarity of formulation.
Previous knowledge
The students should already have followed a basic training in analysis (e.g. as provided in the courses Analyse I (B-KUL-G0N30B) and Analyse II (B-KUL-G0N86B). In particular, this course elaborates Lebesgue's integration theory which is intiated in Analyse II. Moreover, it can be helpful if the student is familiar wth the basic concepts and results from probability theory as treated in, e.g., Kansrekenen en statistiek I (B-KUL-G0Z26A) and Kansrekenen en statistiek II (B-KUL-G0N96B).
Identical courses
This course is identical to the following courses:
G0P63C : Probability and Measure Online
Is included in these courses of study
- Master of Statistics and Data Science (on campus) (Leuven) (Theoretical Statistics and Data Science) 120 ects.
- Courses for Exchange Students Faculty of Science (Leuven)
- Master in de wiskunde (Leuven) 120 ects.
- Master of Mathematics (Leuven) 120 ects.
Activities
4 ects. Probability and Measure (B-KUL-G0P63a)
Content
In this course, measure theory is developed as a general, conceptually elegant and technically efficient integration theory. It is shown how measure theory provides the tools and part of the language for Kolmogorov's formalism for rigorous probability theory. Moreover the measure theoretical language is extended with typical probabilistic concepts (such as independence) and results. Below a general overview of possible themes and subjects is described.
The need for measure theory from integration theory and probabilty theory:
The incompleteness of the Riemann integral, Lebesgue's idea. Kolmogorov's formalism for probability theory
The general Lebesgue integration theory:
Measure spaces. Integration of measurable functions. Convergence theorems (monotone, dominated, Fatou's lemma)
Construction of measure spaces:
Outer measures and Carathéodory's construction. Lebesgue measure, Lebesgue-Stieltjes measures, distribution functions. Comparison between the Lebesgue integral and the Riemann integral.
Kolmogorov's formalism for probability:
Probability spaces. Random variables (distribution, distribution function, expected value). Indepence (for events, random variables and sigma-algebras). Borel-Cantelli lemmas. Tail-sigma-algebras and Kolmogorov's 0-1-law.
Product measure spaces:
Construction (including infinite products of probabilty spaces). Fubini's theorem. Convolutions. Independence and product constructions.
Absolute continuity and singularity:
Radon-Nikodym-derivative (density functions). Lebesgue's decomposition theorem. Conditional expectations.
Lp spaces:
Completeness, Hölder and Minkowski inequalities. Duality.
Convergence of sequences of measures and random variables:
Weak convergence, convergence in distribution, convergence in probabilty, Helly’s selection theorem.
Course material
Lecture notes.
Format: more information
Lectures are integrated with exercise sessions.
Is also included in other courses
2 ects. Probability and Measure: Exercises (B-KUL-G0P64a)
Content
see G0P63a.
Course material
see G0P63a
Format: more information
Exercise sessions are integrated with the lectures.
Is also included in other courses
Evaluation
Evaluation: Probability and Measure (B-KUL-G2P63b)
Explanation
More information will be announced on Toledo.
Information about retaking exams
More information will be announced on Toledo.