School of Mathematics
Module MA2224 - Lebesgue integral
2011-12 (SF Mathematics, SF Two-subject Moderatorship
)
Lecturer: Professor Richard Timoney
Requirements/prerequisites:
prerequisite: MA2223 (or 121)
Duration: Hilary term, 11 weeks
Number of lectures per week: 3 lectures including tutorials per week
Assessment: Tutorial work 15%.
ECTS credits: 5
End-of-year Examination:
This module will be examined jointly with MA2223
in a 3-hour examination in Trinity term,
except that those taking just one of the
two modules will have a 2 hour examination.
However there will be separate grades for MA2223 and MA2224.
Description:
The basics of the theory of the Lebesgue integral and Lebesgue measure
on the real line. Monotone and dominated convergence theorems.
In more detail:
- Countable versus uncountable sets;
inverse images; characteristic functions; boolean algebra for subsets.
-
Algebras of subsets of the real line; length measure on the interval
algebra; finite-additivity, subadditivity and countable-additivity;
outer measure;
Lebesgue
measurable sets;
extension to sigma algebra; Borel sigma algebra.
-
Lebesgue measurable functions; simple functions; integrals for
non-negative functions; limits of measurable functions and the
monotone convergence therorem; Lebesgue integrable functions;
generalisation
of the Riemann integral (for continuous functions on finite closed
intervals).
-
Fatou's lemma; dominated convergence theorem; integrals depending
on a parameter; almost everywhere.
See http://www.maths.tcd.ie/~richardt/MA2224 for additional
information.
Learning
Outcomes:
On successful completion of this module, students will be able to:
- discuss countable sets, characteristic functions and bolean
algebras;
-
state and prove properties of length measure, outer measure and
Lebesgue measure for subsets of the real line and establish
measurability for a range of functions and sets;
-
define the Lebesgue integral on the real line and apply basic results
inclucing convergence theorems.
Jan 15, 2012
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On 15 Jan 2012, 14:01.