By Smirnov V.I.

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**Extra resources for A course of higher mathematics, vol. 2**

**Example text**

42. Obviously (iii) implies (ii)’. Assume that (ii)’ holds. 29. Set Ej = {B ∈ A : B ⊂ Aj } and define fj : Ej → X by fj (B) = µ(B). 29 now gives the result. We consider the semi-variation of set functions µ : A →X with values in a normed space X. 1). 44. For A ∈ A the semi-variation of µ on A is defined to be n µ (A) = sup tj µ(Aj ) : {Aj }nj=1 a partition of A and |tj | ≤ 1 . j=1 We have the following properties of the semi-variation. 7). 45. Let µ : A →X. (i) µ (A) = sup{|x µ| (A) : x ≤ 1}, (ii) sup{ µ(B) : B ⊂ A, B ∈ A} ≤ µ (A) ≤ 2 sup{ µ(B) : B ⊂ A, B ∈ A}.

October 9, 2008 15:36 World Scientific Book - 9in x 6in MasterDoc Multiplier Convergent Series 42 Suppose that µ is bounded with sup{|µ(A)| : A ∈ A } = M < ∞. Let {Aj } ⊂ A be pairwise disjoint. 37, strongly bounded. ∞ j=1 µ(Aj ) = |µ(∪j∈σ Aj )| ≤ M |µ(Aj )| ≤ 2M . 39. 38 shows that if µ : A → R is bounded and finitely additive and {Aj } is pairwise disjoint, then the series ∞ j=1 µ(Aj ) is absolutely convergent. Thus, if µ fails to be countably additive, the series ∞ j=1 µ(Aj ) converges but may fail to converge to the ”proper value”, namely, µ(∪∞ j=1 Aj ).

45. Let µ : A →X. (i) µ (A) = sup{|x µ| (A) : x ≤ 1}, (ii) sup{ µ(B) : B ⊂ A, B ∈ A} ≤ µ (A) ≤ 2 sup{ µ(B) : B ⊂ A, B ∈ A}. , n. Then n n : x ≤ 1} j=1 tj µ(Aj ) = sup{ x , j=1 tj µ(Aj ) n ≤ sup{ j=1 | x , tj µ(Aj ) | : x ≤ 1} n ≤ sup{ j=1 | x , µ(Aj ) | : x ≤ 1} n ≤ sup{ j=1 |x µ| (Aj ) : x ≤ 1} = sup{|x µ| (A) : x ≤ 1}. Therefore, µ (A) ≤ sup{|x µ| (A) : x ≤ 1}. , An } be a partition of A. Then n n j=1 (signx µ(Aj ))x µ(Aj ) j=1 | x , µ(Aj ) | = = ≤ x, n j=1 (signx n j=1 (signx µ(Aj ))µ(Aj ) µ(Aj ))µ(Aj ) ≤ µ (A).

### A course of higher mathematics, vol. 2 by Smirnov V.I.

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