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By Daniel W. Stroock

This variation develops the fundamental idea of Fourier rework. Stroock's process is the only taken initially via Norbert Wiener and the Parseval's formulation, in addition to the Fourier inversion formulation through Hermite services. New routines and suggestions were additional for this variation.

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We have already encountered such a problem when we wrote the equation JL(r2 \ ri ) == JL(r2 ) - JL(ri ) in Theorem 3. 6. Our problem there, and the basic one which we want to discuss here, stems from the difficulty of extending the arithmetic operations to include oo and - oo. Thus, in an attempt to lay all such technical difficulties to rest once and for all, we will spend a little time discussing them here. To begin with, we point out that lR admits a natural metric with which it becomes compact.

Then G U� Gn is open, contains r, and (by Lemma 2. 2) satisfies CX) CX) U( Gn \ rn ) e IG \ rle < 1 I U 8I, we see that every rectangle is 0 Finally, by writing a rectangle I measurable. D Knowing that BRN is closed under countable unions, our next goal is to prove that it is also closed under complementation. 4) holds for I I on BRN . Our proof will turn on an elementary fact about the topology of �N . Recall that a cube Q in �N is a rectangle all of whose sides have the same length. 2. 9 Lemma.

1 7 0, by itsShowintegrals. that an integrable function is determined, up to a set To be more precise, let (E, B, J-L) be a measure space and f and g a pair of functions from L 1 (J-L) . Show that f � g if and only if fr f dJ-L fr g dJ-L for each r E B. Reduce the problem to showing that if E £ 1 (J-L), then J-L ( { < 0} ) 0 if and only if fr dJ-L > 0 for every r E B. 3 cp Convergence of Integrals. == One of the distinct advantages that Lebesgue ' s theory of integration has over Riemann ' s approach is that Lebesgue ' s integral is wonderfully continuous with respect to convergence of integrands.

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