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Unformatted text preview: Analysis 2
Miscellaneous Problems 4
1. Let U be a free (non-principal) ultrafilter on Z ; define µ : P(Z ) → {0, 1} by
µ(A) = 1 when A ∈ U and µ(A) = 0 when A U . Prove that µ is σ-additive iff
U is closed under countable intersections.
2. Let (an )n∈N be a bounded real sequence. Prove that the following conditions are
equivalent:
(i) there exists a subsequence (ank )k∈N converging to b;
(ii) there exists a free u.f. U on N with limU an = b.
3. Let U be a free u.f. on N and (an )n∈N be a bounded real sequence; prove that
lim an ∈ [lim inf an , lim sup an ].
U n n 1 ...
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This note was uploaded on 07/08/2011 for the course MHF 3202 taught by Professor Larson during the Spring '09 term at University of Florida.
- Spring '09
- LARSON
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