16
CHAPTER 2. MATHEMATICAL PRELIMINARIES
2.2 Set Theory
You recall that a set is a well-dened collection of objects. That is, a set is dened if one can
determine whether any given object belongs to the set or not. Hence, the notions of element and
subset of a set. One then denes basic set operations such as union, intersection, complement,
and dierence. The next concept is that of function
f
dened from a set into another set
S
: the function assigns an element
f
(
!
) of
S
to each element
!
of . We designate such a
function by
f
:
!
S
.
If
f
:
!
S
and
B
S
, then
f
1
(
B
) :=
f
!
2
j
f
(
!
)
2
B
g
is the
inverse image
of
B
under
f
(
). You can show that the inverse image commutes with set operations. For instance,
f
1
(
\
i
2
I
B
i
) =
\
i
2
I
f
1
(
B
i
) and
f
1
((
A
n
B
)
[
D
c
) = (
f
1
(
A
)
n
f
1
(
B
))
[
(
f
1
(
D
))
c
.
A set is
countable
if one can enumerate its elements as =
f
!
1
; !
2
; : : :
g
. The rational
numbers are countable, the real numbers are not. A countable union of countable sets is
countable.
For
n
1, let
A
n
and let also
A
. We write
A
n
#
A
if
A
n
+1
A
n
;
8
n
1 and if
\
1
n
=1
A
n
=
A
. We dene
A
n
"
A
similarly. More generally, we say that the sets
A
n
converge to
A
and we write
A
n
!
A
if
[
1
m
=
n
A
m
#
A
and
[
n
m
=1
"
A
.
2.3 Real Numbers
Let
S
<
. We say that
x
2 <
is a
lower bound
of
S
if
x
y
for all
y
2
S
. If
x
2
S
is a lower
bound of
S
, we write
x
= min
S
and we say that
x
is the
minimum
of
S
. The
completeness
axiom
of real numbers states that if a set
S
of real number has a lower bound
x
, then it has a
greatest lower bound
y
and we write
y
= inf
S
and we say that
y
is the
inmum
or the
greatest
lower bound
of
S
. We have the similar denitions for the
maximum
max
S
and the
supremum
sup
S
or
lowest upper bound
. Thus, 0 = min[0
;
1] = inf(0
;
1] but (0
;
1] does not have a minimum
element.
Let
x
2 <
and
x
n
2 <
for
n
1. We say that
x
n
converges to
x
and we write
x
n
!
x
if for
all
>
0 there is some
n
(
) such that
j
x
n
x
j
;
8
n
n
(
). We also write
x
n
! 1
if for all
c
2 <
these is some
n
(
c
) such that
x
n
c
for all
n
n
(
c
).
An important fact is that a nondecreasing sequence that is bounded from above must
converge.
The sequence
f
x
n
; n
1
g
is
Cauchy
if sup
k;m
n
j
x
k
x
m
j !
0 as
n
! 1
. One has
x
n
!
x
with
x
nite if and only if the sequence is Cauchy. For a proof, see Lemma 21.1
2.4 Probability
We review in Appendix A the basic notions of Probability Theory. Here we highlight a few
ideas.
Let =
f
1
;
2
;
3
;
4
g
and assume that each element has probability 1/4. Consider the events
A
=
f
1
;
2
g
; B
=
f
1
;
3
g
; C
=
f
1
;
4
g
. These events are pairwise independent but not mutually
independent. Indeed,
P
(
A
\
B
) =
P
(
A
)
P
(
B
) and similarly for the other pairs of sets, but
P
(
A
\
B
\
C
)
6
=
P
(
A
)
P
(
B
)
P
(
C
). Thus,
P
[
A
j
B
\
C
]
6
=
P
(
A
). The point of this example is
that one should not confuse the statements `
A
and
B
are independent' and `knowing that
B
occurs does not tell us anything about how
A
occurs.'
Consider a probability space
f
;
F
; P
g
with = [0
;
1]
;
F
=
f
[0
;
0
:
5]
;
(0
:
5
;
1]
;
;
;
g
; P
([0
;
0
:
5]) =
0
:
3
:
Let also
X
:
! <
be dened by
X
(
!
) =
!
. Then
X
is not a random variable
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