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Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii) For all &> 0,30- an open set, OE such that m* (OE) ≤ε. 00 (iii) 3G, a Gs-set, G2E such that m* (GE) = 0, (A set G is said to be Gs if G = G₁, i=1 each G is an open set.)

Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii) For all &> 0,30- an open set, OE such that m* (OE) ≤ε. 00 (iii) 3G, a Gs-set, G2E such that m* (GE) = 0, (A set G is said to be Gs if G = G₁, i=1 each G is an open set.)

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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Show that ii implies 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, OE such that m* (0 – E) ≤ ɛ . n i=1 (iii) 3G, a Gs-set, G₂E such that m* (G – E) = 0, (A set G is said to be Gs if G = N G₁, each G is an open set.) (iv) For all ɛ > 0,3 F – a closed set, FCE, such that m* (E – F) ≤ ɛ. 8 i=1 (v) ¨ ³ F, a F-set, F≤E such that m* (E – F) = 0. (A set F is said to be F if F = U F₁, 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, OE such that m* (0 – E) ≤ ɛ . n i=1 (iii) 3G, a Gs-set, G₂E such that m* (G – E) = 0, (A set G is said to be Gs if G = N G₁, each G is an open set.) (iv) For all ɛ > 0,3 F – a closed set, FCE, such that m* (E – F) ≤ ɛ. 8 i=1 (v) ¨ ³ F, a F-set, F≤E such that m* (E – F) = 0. (A set F is said to be F if F = U F₁, each F is a closed set.)
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