Textbook:
The Elements of Integration and Lebesgue Measure
,
R. G. Bartle 1995.
Resources:
- For motivation, read about the
Banach-Tarski Paradox,
or think about what the integral of the
(non Riemann integrable) characteristic function
of the rationals (which are countable) should be.
-
Lecture notes
(updated latex)
prepared by a former student Jennifer Bramwell.
Lecturer:
Shayne Waldron.
Class times:
The class meets on Monday 9 am (in Clock Tower East, Room G10), Thursday 12 pm (in Clock Tower, Room 018), Friday 9 am (in Clock Tower East, Room G10).
Exam: To be announced.
Assignments: There will be about ten assignments.
The assignments will be handed out on Friday, and collected the following
Friday. Marked assignments will be handed back with model solutions.
Assessment: The final grade will be calculated:
30% assignments and 70% final exam, or 100% on the final exam,
whichever is the maximum.
Syllabus:
We will follow the text book quite closely. Below is a rough summary of
the what will be covered in the first 16 lectures.
- Introduction, set theory: cardinality and de Morgan's laws
- The extended reals, and infinite limits of sequences
- The liminf and limsup of a sequence of extended reals,
and their properties.
- sigma algebras, and the example of the Borel sigma algebra
- measurable functions
- sums, products, limits, limsups, etc of measurable functions
- measures, and a proof there is no measure on the power set of the
reals which has the basic properties we would desire
- the measure of expanding and contracting sequences of sets,
and sets of measure zero, including the Cantor set
- Simple function, the approximation of a nonnegative measurable function by an increasing sequence of simple functions
- The integral of a simple nonnegative function
- Integration of nonnegative measurable functions, and the monotone
convergence theorem
- Fatou's Lemma, integration against a nonnegative measurable
function gives a measure.
- The integral of a nonnegative function is zero iff it is zero a.e.
Definition of the integral of f in terms of the integral of its positive and
negative parts and space L of integrable functions.
- Dominated convergence theorem, and the continuous version.
- Norms, and L_p spaces (everything but Minkowskii's inequality
and completeness proved).
- L_p is a Banach space.
- Brief comments about different modes of convergence.
- Brief comments about signed measures and charges, the Jordan decomposition
and the Radon-Nikodym theorem (we may return to this if time permits).
Midsemster break. We now aim to construct the Lebesgue and product measures.
- Overview: outer measures and the Caratheodory magic construction.
Show that a measure on an algebra gives a natural outer measure.
- Show that a measure of length is define on the algebra generated by
the half open intervals.
- Caratheodory magic.
- The construction of Lebesgue measure (or Lebesgue-Stieljes measure)
- More on Lebesgue measure: translation invariance, inner and outer regularity. Haar measure.
- Convex functions and Jensen's inequality (3 lectures).
- Equality in Holder's inequality.
- The represensation of the continuous linear functionals on
L_p(mu), 1