By James Brown, Ruel Churchill
Advanced Variables and purposes, 8E
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When N is closed and discrete, every atomic measure defined by a function a such that a(x) = 0 on eN is called a discrete measure on X (cf. 5). II. Lebesgue measure. For every function f E f (R; C) , there exists a compact interval [a, b) of R outside of which f is zero. 5, Prop. 6), we have II(J) I ::;;; (b - a)lIfll; this shows that f 1-+ I(J) is a measure on R, which is called Lebesgue measure. For every interval J (bounded or not) of R, one similarly calls Lebesgue measure on J the measure f 1-+ fJ f(x) dx, a linear form on f(J; C) (the integral having meaning since there exists a compact interval [a, b) contained in J outside of which f is zero).
Finally, show that there exist filters on E that are convergent for ,%(E) but are not bounded. EI a family of elements of E. ; LX•. H (make use of Exer. 9 a)). b) Generalize to summable families, for the order structure of E, the properties of summable families in commutative topological groups (GT, III, §5, Props. 2, 3 and Th. 2). EI be a family of elements): 0 of E; for the family to be summable, it is necessary and sufficient that the set of finite partial sums SH be bounded above (make use of Exer.
6, show that every archimedean Riesz space of dimension n is isomorphic to the product space R n (cf. §1, Exer. 13 c) Give an example of a totally ordered Riesz space of dimension 2. ~ e». ) a family of positive linear forms on E. (lxl). (lxl) = 0 for all t is an isolated subspace of E (§1, Exer. 21 RIESZ SPACES §2 quotient space E/H; by passage to the quotient, the U, define positive linear forms U, on E/H, and the quotient topology on E/H is defined by the semi-norms U,(I:i:I). b) Show that in the space E the mapping x f-> Ixl is uniformly continuous, and deduce therefrom that if the topology § is Hausdorff then it is compatible with the ordered vector space structure of E.