00 t to an element f of 1/, f being known as the swn of the series. 6 Lemma. A normed vector space is complete if and only if every absolutely convergent series in it is convergent. Ina Banach space the terms in the series Lfn may be rearranged without affecting its sum if the series is absolutely convergent. Proof The only difficulty comes in proving the if part of the first statement; we need to show that if every absolutely convergent series is convergent, then every Cauchy sequence is convergent.
Ba is obviously a measure on Y. If X = IRn, the b-function is obtained. This example reassures us that measures do exist, but the Dirac measure certainly does not yield the usual length in R It is to this case that we now turn. It may be shown that no such measure can be defined on the e-algebra of all subsets of R To find a measure analogous to length on the Borel sets Y of IR, it is natural to start with intervals, say with end points a and b, and to take their measure to be b - a. We may then hope that on combining these basic sets by taking countable unions and so on, we will end up with a measure on Y.