Notes 3 for Honors Probability and Statistics
Ernie Croot
September 12, 2003
1
Measure Theory and Integration
Now that we have a measure on
B
(
R
), namely the BorelLebesgue measure,
we can try to see what measure it assigns to certain sets.
The main tool
we will use is the Monotone Convergence Theorem for Sets in one of the
following two forms:
MCT1.
Suppose that Σ is a
σ
algebra on a set
S
, and that
μ
: Σ
→
[0
,
∞
]
is a measure. If
A
1
, A
2
, ...
are subsets in Σ satisfying
A
1
⊆
A
2
⊆ · · ·
then we
have that if
A
=
∪
j
A
j
,
lim
k
→∞
μ
(
A
k
) =
μ
(
A
)
.
Note: We had proved this before for probability measures, but it holds for
measures in general.
MCT2.
Suppose
S,
Σ
,
and
μ
are as above.
Given
A
1
⊇
A
2
⊇
A
3
· · ·
, all
lying in Σ, let
A
be their intersection. Then, we have
lim
k
→∞
μ
(
A
k
) =
μ
(
A
)
.
Note: This is a homework problem you were asked to solve, except that here
μ
is a general measure, not just a probability measure.
Now, suppose that
μ
is the Lebesgue measure on
B
(
R
). Then, we have
the following two basic observations:
I. For any
a
∈
R
,
μ
(
{
a
}
) = 0. To see this, we apply MCT2: Let
A
j
be
the open interval (
a

1
/j, a
+ 1
/j
). Then,
{
a
}
=
∩
j
A
j
. So,
μ
(
{
a
}
) =
lim
j
→∞
μ
(
A
j
) =
lim
j
→∞
2
j
= 0
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
II. As a corollary of I we get that any countable subset of
R
has measure
0 (why?).
Here is a question that leads to a deeper appreciation of just how special
the Lebesgue measure is:
Question:
Must we have that if
λ
is a measure on
B
(
R
) then
λ
(
{
a
}
) = 0?
It turns out that the answer is NO; that is, there do exist strange measures
on
B
(
R
) which assign singletons nonzero measures. A good example of such
a measure is the following one: Suppose that
A
∈
B
(
R
). Then we have that
λ
(
A
) =
1
,
if 0
∈
A
;
0
,
if 0
6∈
A.
It is easy to check that this is a measure on
B
(
R
), and it is obvious that
λ
(
{
0
}
) = 1.
The natrual progression of our discussion of the Lebesgue measure at
this point would be to define the Lebesgue integral, and then state and prove
various theorems about it. However, to do this task properly would take too
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Statistics, Sets, Probability, measure, Lebesgue measure, Lebesgue integration, Lebesgue, lim µ

Click to edit the document details