Final Exam, Fall 2009, Statistics 778
Answer any FIVE questions. Answers must be supported by clear argu
ments. If you use any theorem, clearly indicate that.
1. Deﬁne symmetric diﬀerence
A
Δ
B
of two sets
A
and
B
.
Show that (
∪
∞
i
=1
A
i
)Δ(
∪
∞
i
=1
B
i
)
⊂ ∪
∞
i
=1
(
A
i
Δ
B
i
).
Let (Ω
,
A
,μ
) be a measure space and
E
be a sub
σ
ﬁeld of
A
. Consider
F
=
{
F
∈
A
:
μ
(
E
Δ
F
) = 0 for some
E
∈
E
}
. Show that
F
is also a
σ
ﬁeld, containing
E
.
Construct an example to argue that
F
may be strictly bigger than
E
.
If
f
is an
F
measurable function, show that there exists an
E
measurable
function
g
such that
f
=
g
a.e. [
μ
]. (Hint: Start with indicator functions.)
[2+4+4+5+5=20]
2. Let
μ
be a LebesgueStiltjes measure with continuous and strictly in
creasing distribution function
F
.
Show that
μ
(
A
) = 0 for any countable set
A
.
If
μ
(
A
) = 0, is
A
necessarily countable? Prove or give a counterexample
with all details.
If
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 Spring '10
 GHOSHAL
 Statistics, Probability theory, Distribution function, probability density function, measure, LebesgueStiltjes measure

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