Physics 129a
Measure Theory
071126 Frank Porter
1
Introduction
The rigorous mathematical underpinning of much of what we have discussed,
such as the construction of a proper Hilbert space of functions, is to be found
in “measure theory”. We thus reFne the concepts in our note on Hilbert
spaces here. It should immediately be stated that the term “measure” refers
to the notion of measuring the “size” of a set. This is the subject of measure
theory. With measure theory, we will Fnd that we can generalize the Riemann
notion of an integral. [This note contains all the essential ideas to complete
the development begun in the note on Hilbert spaces for the construction of
a suitable Hilbert space for quantum mechanics. However, this note is still
under consruction, as there remain gaps in the presentation.]
Let us motivate the discussion by considering the space,
C
2
(
−
1
,
1) of
complexvalued continuous functions on [
−
1
,
1]. We deFne the norm, for any
f
(
x
)
∈
C
2
[
−
1
,
1]:

f

2
=
Z
1
−
1

f
(
x
)

2
dx.
(1)
Consider the following sequence of functions,
f
1
,f
2
,...
,in
C
2
[
−
1
,
1]:
f
n
(
x
)=
⎧
⎪
⎨
⎪
⎩
−
1
−
1
≤
x
≤−
1
/n
,
nt
−
1
/n
≤
x
≤
1
/n
,
11
/n
≤
x
≤
1.
(2)
±ig. 1 illustrates the Frst few of these functions. This set of functions deFnes
a Cauchy sequence, with convergence to the discontinuous function
f
(
x
±
−
1
−
1
≤
x
≤
0,
10
<x
≤
1.
(3)
Since
f
(
x
)
/
∈
C
2
[
−
1
,
1], this space is not complete.
How can we “complete” such a space?
We need to add discontinuous
functions somehow. Can we simply add all piecewsie continuous functions?
Consider the sequence of piecewise continuous functions,
g
1
,g
2
:
g
n
(
x
⎧
⎪
⎨
⎪
⎩
0
−
1
≤
x<
−
1
/n
,
1
−
1
/n
≤
x
≤
1
/n
,
01
/n < x
≤
1.
(4)
1