The Daniell Closure
نویسنده
چکیده
n=1 An = ∅, then lim n→∞ μ(An) = 0. (1.4) Given the ring A of subsets of Ω, a function f : Ω→ R is called A-elementary, if f(Ω) is a finite subset of R and f−1({c} ∈ A for every c ∈ R such that c 6= 0. let E = EA denote the set of all A-elementary functions on E. Provided with the pointwise addition and scalar multiplication of functions, the space F of all functions f : Ω→ R is a vector space over R, and E is a linear subspace of F . Also, in F we have the partial ordering defined by f ≤ g if and only if f(x) ≤ g(x) for all x ∈ Ω, and the functions max(f, g) and min(f, g), defined by (max(f, g))(x) := max(f(x), g(x)) and min(f(x), g(x)) := min(f(x), g(x)) for every x ∈ Ω, are the smallest and largest elements of F which are ≥ f, g and ≤ f, g, respectively. The ring property of A implies that E has the Riesz property f, g ∈ E ⇒ max(f, g) ∈ E and min(f, g) ∈ E. (1.5)
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