Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Sequence and Series
A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.
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Question: Let B be a ring on 2, µ : B → R, be additive. Let A, BE B. (a) Show that u(A) = µ(A\ B) + µ(An B). (b) Show µ(A) + µ(B) = µ(AUB) + µ(An B). Hint: One way to do this is to use (a) and recall that (A\B)(B\A) (AUB) \(An B). (c) Show that 4(AUB) = µ(A)+µ(B) H(ANB) provided that µ(ANB) E R. If N is a nonempty set and BC P(N) is nonempty, then u : B → Ro is called a set function. Definition. Let B be a ring on N and µ : B → R • µ is called additive if µ(AUB) = µ(A) +µ(B) for all disjoint A, B e B. • u is called o-additive if uH An) = EH(An) n=1 n=1 for any sequence (An) of pairwise disjoint sets in B with +, An E B. • µ is called non-negative if µ(A) > 0 for all A e B. • u is called increasing if µ(A) < µ(B) whenever A, B E B and A C B. • u is called subadditive if µ(A U B) < µ(A) + µ(B) for all A, B E B. • µ is called continuous from below if u U An) -lim μ (Α) n00 n=1 for any increasing sequence (An) in B with UnEN An E B.
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Transcribed Image Text:Question: Let B be a ring on 2, µ : B → R, be additive. Let A, BE B. (a) Show that u(A) = µ(A\ B) + µ(An B). (b) Show µ(A) + µ(B) = µ(AUB) + µ(An B). Hint: One way to do this is to use (a) and recall that (A\B)(B\A) (AUB) \(An B). (c) Show that 4(AUB) = µ(A)+µ(B) H(ANB) provided that µ(ANB) E R. If N is a nonempty set and BC P(N) is nonempty, then u : B → Ro is called a set function. Definition. Let B be a ring on N and µ : B → R • µ is called additive if µ(AUB) = µ(A) +µ(B) for all disjoint A, B e B. • u is called o-additive if uH An) = EH(An) n=1 n=1 for any sequence (An) of pairwise disjoint sets in B with +, An E B. • µ is called non-negative if µ(A) > 0 for all A e B. • u is called increasing if µ(A) < µ(B) whenever A, B E B and A C B. • u is called subadditive if µ(A U B) < µ(A) + µ(B) for all A, B E B. • µ is called continuous from below if u U An) -lim μ (Α) n00 n=1 for any increasing sequence (An) in B with UnEN An E B.
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