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Extra resources for An Introduction to Abstract Harmonic Analysis
Proof. We first remark that, if/,. E: U,/,. ~ 0 and L:r II f,. liP 00, then/= f,. lip· For g,. IIP ~ L:i II /• liP by the Minkowski inII and LP E: L:i/i equality, and since g,. lp, by 120. } by any Cauchy sequence in LP. IP < 2-". If g,. - L:: [ ]i+t - j, \ and 39 INTEGRATION + hn = fn L:: I fi+l - fi I, it follows from our first remark that gn and hn E::: LP and that II hn - gn liP < 2-n+2. Moreover, the sequences gn and hn are increasing and decreasing respectively, and if we take/ as, say, limgn, then/ E::: LP and II J- fn liP~ II hn- gn liP< 2-n+2.
Since the integrals ](f) and /(jj0 ) are identical on £2(K) and, in particular, on L, they are identical on DU). This proof is also valid for the more general situation in which J is not necessarily absolutely continuous with respect to I. We simply separate off the set N where g ~ 1, which is /-null as above but not now necessarily ]-null, and restrict f to the complementary set S - N. g', +/ 15C. Theorem. If I is a bounded integral and F is a bounded linear functional on LP(I), 1 ~ p < oo, then ihere exists a unique 42 INTEGRATION function fo E::: L'l (where q = p/(p - 1) if p p = 1) such that II /o llq = II F II and > 1 and q = oo if F(g) = (g,/o) = l(gfo) for every g E::: L".
The axioms of 12A for an elementary integral are readily verified. We first remark that for a functionf(x, y) of the above sort the definition of J = J(J f Jf is actually in terms of the iterated integral: f dx) dy = J(J f dy) dx. If now fn(x, y) 1s a se- quence of elementary functions and /n function of x for every y and therefore ! J 0, then fn /n dx ! 0 as a 0 for every y by the property 12G of the integral in the first measure space S 1 • But then J(J /n dx) dy ! gral in the second space. 0 by the same property of the inteHence J/n dJL !