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Lecture 1. 10.10.2011.
I. MEASURE THEORY AND INTEGRATION
1. Length, Area and Volume.
On the real line, for a b, the length of the interval [a, b] is (dened
to be) b a, and similarly for (a, b), [a, b), (a, b]. In the plane, the area of
a rectangl
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Lecture 2. 10.10.2011.
2. Classes of sets.
We adopt the usual notational convention: lower case letters for points,
or elements of sets; capitals for sets; curly capitals for classes of sets.
We begin with the class O of open sets O. Recall that
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Lecture 3. 14.10.2011.
3. Measures.
Following Lebesgue (1902), our next task is to study the mathematics of
length, area and volume (in Euclidean space of dimensions 1, 2, 3, understood). It turns out that to do this, we actually do much more.
Le
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Lecture 6. 21.10.2011.
4.3. The measure-theoretic integral; the Lebesgue integral.
If f : (, A, ) (R, B (R) is dened on a measure space, f is simple,
f = ci IAi , we dene
f d :=
ci (Ai ).
Although the representation f = ci IAi is not unique, ci (
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Lecture 5. 17.10.2011.
4. The Lebesgue integral.
4.1. The Riemann integral.
We begin with our rst exposure to integration the Sixth Form integral.
To nd the area I under the graph of a bounded function y = f (x) between
x = a and x = b, we partit
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Lecture 8. 24.10.2011.
Properties of the integral (continued).
The next result, Fatous lemma, is due to Pierre FATOU (1878-1929) in
1906.
Theorem (Fatous lemma). (i) If fn are integrable and bounded below
by an integrable function g , fn f a.e.,
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Lecture 4. 17.10.2011.
Measures (continued).
Completion (continued).
Usually (but not always) it is convenient and harmless to complete all
measure spaces in this way. In this course, we shall assume completeness
unless otherwise stated. We then
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Lecture 11. 1.11.2010.
Note on probability spaces.
The example of L10 showing that convergence in pr does not imply a.s.
convergence depends on the existence of non-trivial (Lebesgue-)null sets. In
a purely atomic measure space, there are no non
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Lecture 7. 24.10.2011.
Absolute continuity.
Theorem. If f L(),
f d 0
A
(A) 0).
Proof. Write fn = f if |f | n, 0 otherwise. Then |fn | |f |. So
f d: for all > 0 there exists N with
|f |d <
|fn |d + /2
|fn |d
(n N ).
Then for A A with (A) < /(2N )
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Lecture 10. 1.11.2010.
II. PROBABILITY; CONDITIONAL EXPECTATION
1. Probability spaces.
The mathematical theory of probability can be traced to 1654, to correspondence between PASCAL (1623-1662) and FERMAT (1601-1665). However, the theory remaine
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Lecture 12. 5.11.2010
Then the joint distribution function is given by
F (x1 , . . . , xn ) = P (X1 x1 , . . . , Xn xn ) =
n
P (Xi xi ) =
i=1
n
Fi (xi ),
i=1
where Fi is the distribution function of Xi . The Fi are called the marginal
distributi
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Lecture 13. 7.11.2010
8. The Borel-Cantelli lemmas and the zero-one law.
The following results are due to Borel in 1909, F. P. CANTELLI (19061985) in 1917.
Theorem (Borel-Cantelli lemmas). If An are events, A := lim sup An =
cfw_An i.o.:
(i) If