# Topological Riesz Spaces and Measure Theory

## D. H. Fremlin

Language: English

Pages: 284

ISBN: 0521090318

Format: PDF / Kindle (mobi) / ePub

Measure Theory has played an important part in the development of functional analysis: it has been the source of many examples for functional analysis, including some which have been leading cases for major advances in the general theory, and certain results in measure theory have been applied to prove general results in analysis. Often the ordinary functional analyst finds the language and a style of measure theory a stumbling block to a full understanding of these developments. Dr Fremlin's aim in writing this book is therefore to identify those concepts in measure theory which are most relevant to functional analysis and to integrate them into functional analysis in a way consistent with that subject's structure and habits of thought. This is achieved by approaching measure theory through the properties of Riesz spaces and especially topological Riesz spaces. Thus this book gathers together material which is not readily available elsewhere in a single collection and presents it in a form accessible to the first-year graduate student, whose knowledge of measure theory need not have progressed beyond that of the ordinary lebesgue integral.

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the first two. 17A gives a valuable sufficient condition for greatest lower bounds (and, of course, least upper bounds) to be the same whether taken in the whole space or in a subspace; you will recall that this cannot be taken for granted [1XB]. 17B is an important general theorem on the extension of order-continuous increasing linear maps. 17A Lemma If £ is a Riesz space and F is a locally order-dense Riesz subspace of E, then whenever A is a non-empty subset of F and x is the greatest lower

65 and again in Chapter 8. Proposition Let £ be a Riesz space and % a linear space topology on £ such that order-bounded sets are bounded. Then the following are equivalent: (i) whenever <#n)weN is a disjoint sequence in £+ which is bounded above, neH -> 0; (ii) whenever 0 cz A\ in £, and A is bounded above, ^(A\) is Cauchy. Proof (a) Suppose that condition (i) holds. Let us say that a sequence (x^)ie^ has property Pr if it is bounded above and mfieJxi = 0 whenever « / c N has r +1 members.

The locally solid ones are of course dealt with by 24Lf. Of course there are many important examples of Lebesgue topologies. Some are given in 1XD, 1XF, 26B, 2XB, 2XC, 2XF, 2XG, 63K, 6XF, 6X1 and 82H. *The name 'Lebesgue topology' is suggested by 24Lg, which is an abstract version of Lebesgue's Dominated Convergence Theorem [63Md]. It corresponds to 'condition A(ii)' of LUXEMBURG & ZAANEN B.F.S. [note x, § 33]. NAKANO L.T. [§ 6] gives a definition of' continuous 60 LEBESGUE TOPOLOGIES [24

element of lx{X)\ it follows that the duality between /°°(X) and induces a linear space isomorphism between /°°(X)X and V-{X). (c) Finally, we see that for xeP-(X)9 x^Oo x(t) ^ 0 V teX o (x,et) ^ 0 V teX so that the ordering on P-(X) induced by its embedding in /°°(X)X is correct, and the isomorphism is a Riesz space isomorphism. Q (The argument above is of course a particularly easy special case of thatin65A.) Hence l\X) is perfect [33A, 33F]. 2XG The space co(X) Let X be any non-empty set.

so 6^ c a0 [42Ed] and 6^ e 21 for each i < n. Thus y e 5(21); as x and ?/ are arbitrary, 5(21) is a solid linear subspace of 5(93). So \x\ A \y\ e5(21) for every a;e5(21) and ye5(93). By continuity, \x\ A \y\ belongs to the closure of 5(21), which is L°°(2l), for every xeL™{%) and yeL™08); so L°°(2l) is solid in L°°(93). (c) Let n: 21 -> 93 be the canonical map. Since n is onto, n: 5(21) -> 5(93) and n: L°°(2l) -> L«(») are onto [45Cb]; as TT is a Riesz homomorphism, 5(93) s 5(2l)/£, L°°(93) s