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) Showµ(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) ER.If N is a nonempty set and BC P(N) is nonempty, then u : B → Ro iscalled 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 ifuH An) = EH(An)n=1n=1for 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 ifu U An)-lim μ (Α)n00n=1for any increasing sequence (An) in B with UnEN An E B.

Name: Question:Let B be a ring on 2, µ : B → R, be additive.Let A, BE B.(a)
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: 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) Showµ(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)