Economics 245A
Notes for Measure Theory Lecture
Axiomatic Approach
The axiomatic approach introduced by Kolmogorov starts with a set of axioms, as
do all axiomatic approaches, that are taken to be obvious. The axioms must be: (1)
complete, that is all theorems can be proven from the axioms; (2) consistent, that is
the axioms do not contradict each other; and (3) nonredundant, that is no axiom can
be derived from the other axioms.
Technical Detail for Probability Axiom (iii)
The statement of probability axiom
(iii)
that appears in most textbooks is:
If
A
1
; : : : ; A
n
are disjoint sets then
P
(
[
n
i
=1
A
i
) =
n
X
i
=1
P
(
A
i
)
:
(0.1)
Probability axiom
(iii)
given in class refers to the quantity
[
1
i
=1
A
i
, so we must show
that (0.1) continues to hold for
n
=
1
. To verify, let
A
=
[
1
i
=1
A
i
,
B
n
=
[
n
i
=1
A
i
, and
let
C
n
=
A
°
B
n
:
Then
f
C
n
g
1
n
=1
is a monotone decreasing sequence and
lim
n
!1
C
n
=
\
1
n
=1
C
n
=
;
:
For any value of
n
:
P
(
A
) =
n
X
i
=1
P
(
A
i
) +
P
(
C
n
)
;
so we must prove that
P
(
C
n
)
!
0 as
n
! 1
.
Proof: Method, proof by contradiction. Assume that there exists some
° >
0 such
that
P
(
C
n
)
> °
for all
n
. Let
v
(
s
) be the largest value of
n
for which
s
2
C
n
:
Then
P
(
f
s
:
v
(
s
)
> n
g
) =
P
(
C
n
+1
)
;
because
C
n
+1
consists of all events for which
v
(
s
)
> n
. Yet any element
s
of the sample
space belongs to only °nitely many
C
n
because
\
1
n
=1
C
n
=
;
.
Because any element
belongs to only °nitely many
C
n
it is not possible for
P
(
C
n
)
> °
for all
n
.
Hence
P
(
C
n
)
!
0 as
n
! 1
.
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 Fall '08
 Staff
 Economics, Probability, Probability theory, CN

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