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Lim = fl then U fl k>O n0 n>n0 and this implies the measurability of h. Suppose I is a continuous real-valued function defined on a measure space (s, which is also a topological space. In the most interesting topological measure spaces such functions will also be measurable but they need not always be. Consider, for example, the unit interval [0, ii with the ordinary topology. This is a Remark. compact Hausdorff space. Let I consist of countable sets and 22 NOTES ON MEASURE AND INTEGRATION their complements.
For example consider the subset B C R x R (R is the reals with ordinary Lebesgue measure (), B = axN where a is anyreal number and N is the nonmeasurable subset of [0, iii constructed at the conclusion of Chapx v)*(B) = 0 but from Theorem 1 it follows that ter I. ) B As a consequence of the preceding statements it now becomes desirable to know whether or not the theorems we have just proved hold for the completion of (x x Y, x x v) and therefore (in case and 'Y are the Lebesgue or Borel sets) for the Lebesgue sets of X x Y.
It is not x 21?. For each x and y the sets F difficult to show that F c and each consist of a single point and so Example. Consider I 49 THE THEOREMS OF' FUBINI fa(E)czx = while JR I5Ida = 0. We now apply the results of Theorem 1 to define product mea- sure. We shall assume in the remainder of the chapter that ,z and v are a-finite. Definition. Let X x F, X = x %1 be as before. 4 define v(E)d1t = It follows from the monotone convergence theorem that is a measure. , sets. is a collection of pairwise disjoint measurable 50 NOTES ON MEASURE AND INTEGRATION LEMMA.