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Let En, n = 1, 2,... be a sequence of measurable subsets of R. (a) If En ≤ En+1 for n = 1, 2, then show that m(Ux=1 En) = = lim m(En). n→∞ (b) If En+1 ≤ En for n = 1,2,.. and m(E₁) < ∞ then show that m(En) = lim m(En). n→∞ Further construct an example to show that this may fail if m(E₁) : = ∞ (the condition is necessary).

Let En, n = 1, 2,... be a sequence of measurable subsets of R. (a) If En ≤ En+1 for n = 1, 2, then show that m(Ux=1 En) = = lim m(En). n→∞ (b) If En+1 ≤ En for n = 1,2,.. and m(E₁) < ∞ then show that m(En) = lim m(En). n→∞ Further construct an example to show that this may fail if m(E₁) : = ∞ (the condition is necessary).

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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Let En, n = 1, 2,... be a sequence of measurable subsets of R. (a) If En ≤ En+1 for n = 1, 2, then show that 1,2, (b) If En+1 ≤ En for n = m(Ux=1 En) n= = lim m(En). N→∞ 1,2,.. 2,... and m(E₁) < ∞ then show that m (n=1 En) lim m(En). N→∞ = Further construct an example to show that this may fail if m(E₁) = ∞ (the condition is necessary).
Transcribed Image Text:Let En, n = 1, 2,... be a sequence of measurable subsets of R. (a) If En ≤ En+1 for n = 1, 2, then show that 1,2, (b) If En+1 ≤ En for n = m(Ux=1 En) n= = lim m(En). N→∞ 1,2,.. 2,... and m(E₁) < ∞ then show that m (n=1 En) lim m(En). N→∞ = Further construct an example to show that this may fail if m(E₁) = ∞ (the condition is necessary).
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