On Quaternionic Measures
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Advances in Applied Clifford Algebras
On Quaternionic Measures M. E. Luna-Elizarrar´as∗ , A. Pogorui, M. Shapiro and T. Kolomiiets Abstract. In this paper we consider some basic properties of quaternionic measures. Mathematics Subject Classification. Primary 28A10; Secondary 28A33, 30G35. Keywords. Quaternions, Measures.
1. Introduction The notion of a measure is one of the most fundamental objects in mathematics and it would be superfluous to talk much about this. We present now a few lines only in order to explain what we are going to do in the paper, for more details the reader is referred, for instance, to the book of Halmos [10], but for many other sources as well. Let X be a non-empty set and let M be a σ-algebra of subsets of X. A measure (sometimes called a positive measure) is a function μ defined on the measurable space (X, M) whose range is in [0, ∞] =: R+ and which is countably additive, i.e., if {Ai } is a disjoint countable family of elements of M then ∞ ∞ Ai = μ (Ai ) . (1.1) μ i=1
i=1
This definition includes tacitly that the series on the right-hand side converges to a non-negative number or to ∞. This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29–August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen. ∗ Corresponding
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M. E. Luna-Elizarrar´ as et al.
Adv. Appl. Clifford Algebras
We assume that there exists at least one A ∈ M for which μ(A) < ∞. This excludes the trivial situation of the measure identically equal to ∞. Some important properties are: (a) μ(∅) = 0. (b) Any measure is finite additive, i.e., (1.1) holds for a finite number of pair-wise disjoint elements of M. (c) Any measure is monotone: if A, B are in M and A ⊂ B then μ(A) ≤ μ(B). ∞ (d) If {An }n∈N ⊂ M, A = n=1 An , A1 ⊂ A2 ⊂ . . . ⊂ An . . ., then μ(An ) −→ μ(A) as n −→ ∞. ∞ (e) If {An }n∈N ⊂ M, A1 ⊃ A2 ⊃ . . . ⊃ An . . ., A = n=1 An , μ(A1 ) < ∞, then μ(An ) −→ μ(A) as n −→ ∞. Definition 1.1. A measure on a measurable space (X, M) is called σ-finite if there exists a collection of sets {An , n ∈ N} ⊂ M such that ∪∞ n=1 An = X and for each n ≥ 1 it holds that μ (An ) < ∞. Let us recall a notion of a signed measure or charge: Definition 1.2. A signed measure (or a charge) on a measurable space (X, M) is a function λ : M → R ∪ {−∞, ∞}
(1.2)
such that λ (∅) = 0 and λ is countably additive. The origin of the notion of the measure explains why it takes just nonnegative values. At the same time the question arises: can the measure be complex-valued? A complex measure ω is a complex-valued countably additive function defined on M. A good source of basic information may be Chapter 6 of the book Rudin [15]. In accordance with the definition if ω is identically zero then ω is a positive measure. A positive measure is allowed to have +∞ as its value; but it is proved that a complex measure μ has as its values the complex numbers only: any μ(E) is in C. The real measures are defined as σ-additive real-valued functions and the
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