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By Stefan Hildebrandt

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E), with K>. compact in X>.. 9, Prop. 15). 4, Prop. 6). We shall say that a subset H of X(X; E) is strictly compact if it is compact and if there exists a compact subset K of X such that H C X(X, K; E). It follows at once from Proposition 2 that if X is a paracompact locally compact space and if E is Hausdorff, then every compact set in X(X; E) is strictly compact. 9, Def. 5). 4 2 §1 MEASURES spaces X (not paracompact) such that there exist sets in £(X; R) that are compact but not strictly compact (Exercises 3 and 4).

L, g. L2) = g . Ll + g . L2 , (gl g2) . L = gl . (g2 . L) . L) 1---+ g. L and with its additive structure, is thus a module over the ring 't&"(X;C) . 11 5. Real measures. Positive measures Let X be a locally compact space. 1, Prop. 1). to(fz). to(fz) is a (complex) measure on X. Thus, every measure on X may be identified with its restriction to £(X; R).

Let L be a positive linear form on a Riesz space E. In order that a positive linear form M on E belong to the band generated by L in n, it is necessary and sufficient that, for every x ~ 0 in E and every number, E > 0, there exist a number 8 > 0 such that the relations o ~ y ~ x and L(y) ~ 8 imply M(y) ~ E. Let us first show that the condition is necessary. 5, Cor. of Prop. 6) M = sup (inf(nL, M)) . n If one sets Un = M - inf(nL, M), Un is thus a positive linear form on E and inf Un = 0 in n; consequently n (Lemma) Un(x) tends to 0 as n tends to infinity, and there exists an n such that Un(x) ~ E/2.

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