Symmetry And Group

Harmonic Analysis on Semigroups by C. Berg

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By C. Berg

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F, g, h E C+(X) such that h ~ f T+(f) + g. h(x)f(x) h'(x):= { ~(x) + g(x) h(x)g(x) ~(x) h"(x):= + g(x) { then h', h" E C+(X), h' ~ 00, f E and for f, g + C +(X). E C +(X) we have T+(g). f(x) = g(x) = 0, f, h" ~ g and h' T(h) ~ T+(f) + h" = h, implying + T+(g) so that finally T+ is additive on C +(X). We put T- := T+ - T which also is additive, nonnegative and positively homogeneous on C + (X). 3 there are two Radon measures J1b J12 on X such that 41 §2. 1, := J1l - J12 so that T = ~. 3 we also get immediately that there is only one signed Radon measure J1 with this property.

11. Exercise. Let X be locally compact. For a net (Ji(l) on M +(X) and Ji E M + (X) the following conditions are equivalent: (i) Ji(l ~ Ji vaguely; (ii) lim sup Ji(l(K) ~ Ji(K) for each compact K ~ X and lim inf Ji(l(G) ~ Ji(G) for all relatively compact open sets G ~ X; (iii) lim Ji(l(B) = Ji(B) for all relatively compact Borel sets B ~ X such that Ji(oB) = O. 12. Exercise. Show that the set of all Radon measures on a locally compact space taking only values in No = {O, 1, 2, ... , oo} is vaguely closed.

0, 00]. The extension to the case where Il and v are a-finite is completely straightforward and therefore omitted. 13. It is, of course, a natural question to ask if equality in (12) holds for more general functions than just nonnegative lower semicontinuous ones. The following example shows that one cannot, in general, hope for too much. e. every subset is open in Y). e. v(B) = card(B) for all B £; Y. 12 may be applied. The diagonal L\:= {(x, x)IO ~ x ~ I} is closed in X x Y, hencef := 1A is a bounded nonnegative upper semicontinuous function, in particular fis Borel measurable.

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