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
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 con
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
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 (
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
(
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
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 nat
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
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 br
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 identif