TL;DRAbstract
Suppose that we are given a set X, a Riesz subspace E of Rx, and an increasing linear functional f : E → R. My object in this chapter is to discuss conditions under which f is an ‘integral’, that is, when there is a measure µ on X such that ∫ xdµ exists and is equal to fx for every x ∈ E. A necessary, and nearly sufficient, condition is that f should be ‘sequentially smooth’ [71B–71G]. Further conditions on f and E lead, of course, to stronger results [§§ 72, 73].
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Suppose that we are given a set X, a Riesz subspace E of Rx, and an increasing linear functional f : E → R. My object in this chapter is to discuss conditions under which f is an ‘integral’, that is, when there is a measure µ on X such that ∫ xdµ exists and is equal to fx for every x ∈ E. A necessary, and nearly sufficient, condition is that f should be ‘sequentially smooth’ [71B–71G]. Further conditions on f and E lead, of course, to stronger results [§§ 72, 73].
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