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Math 8601. October 6, 2010. Midterm Exam 1. Problems and Solutions.
Problem 1.
Let (
X,
M
,μ
) be a measure space with
μ
(
X
)
<
∞
. Show that for arbitrary
A,B,C
∈ M
,
we have

μ
(
A
∩
B
)

μ
(
A
∩
C
)
 ≤
μ
(
B
Δ
C
)
.
where
B
Δ
C
:= (
B
\
C
)
∪
(
C
\
B
) = (
B
∪
C
)
\
(
B
∩
C
)  the
symmetric diﬀerence
of
B
and
C
.
Proof.
One can use the triangle inequality for the “distance”
ρ
(
A,B
) :=
μ
(
A
Δ
B
),
This property simply means that
ρ
(
A,B
) :=
μ
(
A
Δ
B
) satisﬁes the triangle inequality, which is Problem #5
in HW #2. It is easy to see that
B
0
:=
A
∩
B
and
C
0
:=
A
∩
C
satisfy (
B
0
Δ
C
0
) =
A
∩
(
B
Δ
C
)
⊂
(
B
Δ
C
). Then

μ
(
A
∩
B
)

μ
(
A
∩
C
)

=

μ
(
B
0
)

μ
(
C
0
)

=

ρ
(
B
0
,
∅
)

ρ
(
C
0
,
∅
)
 ≤
ρ
(
B
0
,C
0
) =
μ
(
B
0
Δ
C
0
)
≤
μ
(
B
Δ
C
)
.
Alternatively, one can note that
μ
(
A
∩
B
) =
μ
(
A
∩
B
∩
C
) +
μ
(
A
∩
B
∩
C
c
)
,
μ
(
A
∩
C
) =
μ
(
A
∩
B
∩
C
) +
μ
(
A
∩
B
c
∩
C
)
,
hence

μ
(
A
∩
B
)

μ
(
A
∩
C
)
 ≤
μ
(
A
∩
B
∩
C
c
) +
μ
(
A
∩
B
c
∩
C
)
≤
μ
(
B
∩
C
c
) +
μ
(
B
c
∩
C
) =
μ
(
B
Δ
C
)
.
Problem 2.
For
A, B
⊂
R
1
, deﬁne the algebraic sum
A
+
B
=
{
x
∈
R
1
:
x
=
a
+
b
for some
a
∈
A, b
∈
B
}
.
Either prove or give a counterexample for each of the following statements.
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 Fall '08
 Staff
 Math

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