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MATH 587
Assignment 1
Due October 5, 2004
(1) Let Ω be a countable set,
F
=
P
(Ω). Let
p
(
ω
)
,ω
∈
Ω be nonnegative numbers. Show that
P
(
A
)=
∑
ω
∈
A
p
(
ω
) defnes a
σ
fnite measure on
F
.
(2) Billingsley (page 41) defnes a class
C
oF subsets oF Ω to be a
λ
system iF
1’. Ω
∈C
,
2’.
C
is closed under complementation,
3’.
C
is closed under countable unions oF pairwise disjoint members oF
C
(and so is also closed under
fnite disjoint unions).
Show that this defnition coincides with that oF a
d
system.
(3) Prove the Monotone Class Theorem: Let
A
be an algebra. Then
σ
(
A
)=
M
(
A
). (i.e. the
σ
algebra
generated by
A
coincides with the monotone class generated by
A
.
(4) Let (Ω
,
F
,µ
)bea
σ
fnite measure space. A set
E
∈F
is an
atom
oF
µ
iF
µ
(
E
)
>
0 and iF
µ
(
F
)=0or
µ
(
F
)=
µ
(
E
) whenever
F
∈F
and
F
⊂
E
. Show that
(a) two atoms
A
and
B
are either a.s. disjoint, i.e.
µ
(
A
∩
B
) = 0, or a.s. coincide, i.e.
µ
[
A
4
B
]=0.
(b) iF
A
is an atom, and iF
A
⊂
C
∪
D
where
C
∩
D
=
∅
, then either
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This note was uploaded on 01/25/2011 for the course STAT 235a at Stanford.
 '07
 RomanVershynin
 Probability

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