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Analysis 2
Miscellaneous Problems 2
1. Let
C
be a collection of subsets of the set
Ω
. For each
A
∈
σ
(
C
) show that there
is a countable subcollection
C
A
of
C
such that
A
∈
σ
(
C
A
).
2. Let (
Ω
,
F
) be a measurable space on which (
μ
n
:
n
∈
N
) is an increasing
sequence of measures. Show that a measure
μ
on (
Ω
,
F
) is defined by the rule
A
∈ F ⇒
μ
(
A
)
=
lim
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Unformatted text preview: μ n ( A ) . 3. Let ( Ω , F , μ ) be a finite measure space; for convenience, let ∅ be the only null set. Show that the rule A , B ∈ F ⇒ d ( A , B ) = μ ( A 4 B ) defines a metric d on F . Verify that the operations of union and intersection are uniformly continuous. 1...
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- Spring '09
- LARSON
- Topological space, measure, finite measure space, countable subcollection CA
-
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