By Daniel A. Murray

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**Additional resources for A first course in infinitesimal calculus**

**Example text**

We denote by L' the vector space of d simple functions. If W is a lattice of sets then L' is identical with the class L-q of 9 simple functions, where 9 is the ring generated by 4, because 9 consists of unions of finitely many disjoint sets of the form A\B with A and B in 4. Further, if R is a family closed under intersection and d is the lattice generated by R then 0 = L' = L. We omit the straightforward proof of this fact. , aXA + bxB = aXA\B + (a + b)xAnB + bxB\A). Further, if f and g belong to L`, then we may suppose f = E k 1 ak xck and g = E k=1 bk xCk for some disjoint finite sequence {C1, C2, ...

Thus 1 I (f) I < I I f L (It would be more precise to call 111 the "norm induced by I", and . ) The L 1 norm is unfortunately not always a norm. f 11 = It 1 11 f 11 for all f in E and all scalars , and (iii) for each f in E, if II f II = 0 then f = 0. The L 1 norm has properties (i) and (ii) but, in general, may fail to satisfy (iii). It should properly be called a semi-norm or a pseudo-norm, but we follow time honored usage in calling it a norm. If is a semi-norm for a vector space E then II f - g11 is the II 11 norm distance from f to g, and the function (f, g) H 11 f - g11 is the norm semi-metric.

The (5-ring generated by the family of compact subsets of R is the Borel 6-ring Ra (D) for R. The family of bounded subsets is a (5-ring that contains the family of compact sets and hence contains 41(R). A mea- sure on W(R) is a Borel measure for R. The family V (R) is also generated by the class of closed intervals, and each length function is a pre-measure, so the minimal measure extending a length function is a Borel measure. We show that every Borel measure can be obtained in this way. PROPOSITION The minimal measure extending a length function is a Borel measure for R, and every Borel measure is such an extension.