30
RICHARD B. MELROSE
5. Hilbert space
We have shown that Lp (X, ) is a Banach space a complete normed
space. I shall next discuss the class of Hilbert spaces, a special class of
Banach spaces, of which L2 (X, ) is a standard example, in which the
norm ar
LECTURE NOTES FOR 18.155, FALL 2004
19
4. Integration
The ()-integral of a non-negative simple function is by denition
f d =
ai (Y Ei ) , Y M .
(4.1)
Y
i
Here the convention is that if (Y Ei ) = but ai = 0 then ai (Y
Ei ) = 0. Clearly this integral takes
16
RICHARD B. MELROSE
3. Measureability of functions
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
f 1 (E ) M E N .
(3.1)
Notice
10
RICHARD B. MELROSE
2. Measures and -algebras
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 disjoin
2
RICHARD B. MELROSE
1. Continuous functions
A the b eginning 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 rst try to set the context.
One basic notion I assume you are reas