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Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii) For all ɛ> 0,30 - an open set, 02 E such that m' (O – E) < . (iii) 3G, a Gs-set , G2 E such that m* (G – E) = 0, (A set G is said to be Gs if G = N G;, i=1 each G is an open set.) * (E

Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii) For all ɛ> 0,30 - an open set, 02 E such that m' (O – E) < . (iii) 3G, a Gs-set , G2 E such that m* (G – E) = 0, (A set G is said to be Gs if G = N G;, i=1 each G is an open set.) * (E

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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Prove (ii) => (iii).

Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii) For all ɛ > 0,30 – an open set, 02 E such that m* (0 – E) <E . (iii) 3 G, a Gs-set » G 2 E such that m* (G – E) = 0, (A set G is said to be G; if G = N G;, а i=1 each G; is an open set.) (iv) For all ɛ > 0, 3 F - a closed set, FC E, such that m* (E – F) < E . (v) 3 F, a F,- set, F CE such that m* (E – F) = 0. (A set F is said to be F, if F = U F, %3D i=1 each F is a closed set.)
Transcribed Image Text:Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii) For all ɛ > 0,30 – an open set, 02 E such that m* (0 – E) <E . (iii) 3 G, a Gs-set » G 2 E such that m* (G – E) = 0, (A set G is said to be G; if G = N G;, а i=1 each G; is an open set.) (iv) For all ɛ > 0, 3 F - a closed set, FC E, such that m* (E – F) < E . (v) 3 F, a F,- set, F CE such that m* (E – F) = 0. (A set F is said to be F, if F = U F, %3D i=1 each F is a closed set.)
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