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A measure space (X, μ) is said to be separable if there is a countable family of measurable subsets {E}_1 so that if E is any measurable set of finite measure, then μ(ΕΔΕn) → 0 as k → ∞, for an appropriate sequence {n} which depends upon E. Here AAB denotes the symmetric difference of the sets A and B, that is, AAB = (A − B) U (B − A). (a) Prove that Rd is separable. (b) The space LP = LP(X, ) is separable if there exists a countable collection

A measure space (X, μ) is said to be separable if there is a countable family of measurable subsets {E}_1 so that if E is any measurable set of finite measure, then μ(ΕΔΕn) → 0 as k → ∞, for an appropriate sequence {n} which depends upon E. Here AAB denotes the symmetric difference of the sets A and B, that is, AAB = (A − B) U (B − A). (a) Prove that Rd is separable. (b) The space LP = LP(X, ) is separable if there exists a countable collection

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A measure space (X, μ) is said to be separable if there is a countable family of measurable subsets {Ek}_1 so that if E is any measurable set of finite measure, then µ(E^Enk) →0 as k→ ∞, for an appropriate sequence {n} which depends upon E. Here AAB denotes the symmetric difference of the sets A and B, that is, AAB = (A − B) U (B − A). (a) Prove that Rd is separable. (b) The space Lº LP(X, ) is separable if there exists a countable collection of elements {f}=₁ in Lº that is dense in the Lº norm. Prove that if the measure space (X, µ) is separable, then LP is separable when 1 ≤ p<∞. =
Transcribed Image Text:A measure space (X, μ) is said to be separable if there is a countable family of measurable subsets {Ek}_1 so that if E is any measurable set of finite measure, then µ(E^Enk) →0 as k→ ∞, for an appropriate sequence {n} which depends upon E. Here AAB denotes the symmetric difference of the sets A and B, that is, AAB = (A − B) U (B − A). (a) Prove that Rd is separable. (b) The space Lº LP(X, ) is separable if there exists a countable collection of elements {f}=₁ in Lº that is dense in the Lº norm. Prove that if the measure space (X, µ) is separable, then LP is separable when 1 ≤ p<∞. =
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