By Alan L. T. Paterson

ISBN-10: 0821809857

ISBN-13: 9780821809853

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Thus, there exists a unique measure μ ˆ on (E, E) such that L(f ) = μ ˆf for every f in E+ . We have μ ˆ = ν, since A ∈ E. 5 is another convenient tool for creating new measures from the old. 8 ν(dx) = μ(dx) p(x), x ∈ E, once it is understood that μ and ν are measures on (E, E) and that p is positive E-measurable. Heuristically, we may think of μ(dx) as the amount of mass put by μ on an “inﬁnitesimal neighborhood” dx of the point x, and similarly of ν(dx). 8 takes on the meaning that p(x) is the mass density, at x, of the measure ν with respect to μ.

Thus, if f and g are of bounded variation on [s, t], then so are f + g and f − g. e) The function f is of bounded variation on [s, t] if and only if f = g − h for some real-valued positive functions g and h that are increasing on [s, t]. Hint: To show the necessity, deﬁne g(r) and h(r) for r in (s, t] by 2g(r) = Vf (s, r) + f (r) + f (s), 2h(r) = Vf (s, r) − f (r) + f (s) and show that g and h are increasing and f = g − h. ´ The class of functions f for which Lebesgue-Stieltjes integrals g df are deﬁned is the class of f that are of bounded variation over bounded intervals.

We shall use such versions without further ado whenever the need drives us. 3 deﬁnes the integral μf for every f in E+ . Thus, in eﬀect, integration extends the domain of μ from the measurable sets (identiﬁed with their indicator functions) to the space E+ of all positive measurable functions (and beyond), and hence we may regard μ as the mapping f → μf from E+ ¯ + . 8. We end this section with the following very useful converse. 28 Measure and Integration Chap. 21 Theorem. Let (E, E) be a measurable space.

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