Lecture 2 Notes: Outer Measures
An outer measure such as is a rather crude object since, even if
the Ai are disjoint, there is generally strict inequality in (1.14). It
turns out to be unreasonable to expect equality in (1.14), for
disjoint unions, for a
Lecture 1 Notes: Intro to Course
A the beginning I want to remind you of things I think you already
know and then go on to show the direction the course will be
taking. Let me first try to set the context.
One basic notion I assume you are reasonably fami
Lecture 4 Notes:
Integral
Differences
The ()-integral of a non-negative simple function is by definition
ai(Y Ei) , Y M .
(4.1)
Y f d = i
Here the convention is that if (Y Ei ) = but ai = 0 then ai(Y Ei)
= 0. Clearly this integral takes values in [0, ]. M
Lecture 5 Notes: Banach Space
p
We have shown that L (X, ) is a Banach space a complete normed
space. I shall next discuss the class of Hilbert spaces, a special class of
2
Banach spaces, of which L (X, ) is a standard example, in which the
norm arises fr
Lecture 3 Notes: Power Sets
Suppose that M is a -algebra on a set X 4 and N is a -algebra on
another set Y. A map f : X Y is said to be measurable with respect to
these given -algebras on X and Y if
(3.1)
f 1(E) M E N .
Notice how similar this is to one o
Lecture 7 Notes: Metric
Topology
A good first reference for distributions is [2], [4] gives a more exhaustive treatment.
The complete metric topology on S(Rn) is described above. Next I want
to try to convice you that elements of its dual space S(Rn), hav
Lecture 10 Notes: Sobolev Spaces
The properties of Sobolev spaces are briefly discussed above. If m is a
positive integer then uHm(Rn) means that u has up to m derivatives in
L2 (Rn). The question naturally arises as to the sense in which these
weak deriv
Lecture 9 Notes: Inverse Functions
It is shown above that the Fourier transform satisfies the identity
(9.1)
(0) = (2)n
() d S(Rn) .
Rn
If y Rn and S(Rn ) set (x) = (x + y). The translation-invariance of
Lebesgue measure shows that
() =e
ix
(x + y) dx
= e
Lecture 6 Notes: Lebesgue Measures
So far we have largely been dealing with integration. One thing we
have seen is that, by considering dual spaces, we can think of functions as
functionals. Let me briefly review this idea.
Consider the unit ball in Rn ,
Lecture 8 Notes: Convolutions
We have defined an inclusion map
(8.1)
n
Rn (x)(x) dx S(R ).
S(Rn ) u S(Rn), u() =
This allows us to think of S(Rn ) as a subspace of S(Rn); that is we
habitually identify u with . We can do this because we know (8.1) to be
i