By Roger Godement
Research quantity IV introduces the reader to sensible research (integration, Hilbert areas, harmonic research in crew idea) and to the tools of the idea of modular features (theta and L sequence, elliptic features, use of the Lie algebra of SL2). As in volumes I to III, the inimitable kind of the writer is recognizable right here too, not just as a result of his refusal to put in writing within the compact type used these days in lots of textbooks. the 1st half (Integration), a smart mix of arithmetic stated to be 'modern' and 'classical', is universally precious while the second one half leads the reader in the direction of a truly energetic and really expert box of study, with potentially wide generalizations.
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Extra resources for Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext)
The sum of the measures of the strictly open intervals equals 1/3 + 2/32 + 22 /33 + . . = 1 , and as they are pairwise disjoint, µ(K − C) = 1, and so µ(C) = 0. Lemma 1. A set A ⊂ X is integrable if and only if, for all r > 0, there is an open set U and a compact set K such that K⊂A⊂U & µ(U − K) < r . Every integrable set is the union of a null set and a countable family of compact sets. By lemma 4 and n◦ 4, the condition is sufficient since the characteristic functions of K and U are respectively usc and lsc.
6) µ(f ) = lim µ(fn ) . Obvious. Theorem 6. Let (fn )be a sequence of functions in Lp such that +∞. Then |fn (x)| < +∞ ae. 8) Np (fn ) < fn (x) ae. lim Np (f − f1 − . . − fn ) = 0 , µ(f ) = µ(fn ) if p = 1 . This is theorem 5 applied to functions in Lp . By lemma 2, the (class of the) limit function f is still in Lp and (8) follows from (4) and (7). Corollary. Let fn (x) be a sequence of positive integrable functions. The sum of the series is integrable if and only if fn (x)dµ(x) < +∞ . The series then converges almost everywhere and in the space L1 , and dµ(x).
On the other hand, S is everywhere dense in L2 (R) because any function f ∈ L(R) can be approximated using C ∞ functions zero outside a fixed compact set (Chap. V, n◦ 27, theorem 26), which implies convergence in all Lp spaces. Thus the Fourier transform extends to L2 (R) and the extension is unique. The same would be the case if R were replaced with T: for every f ∈ L2 (T), there is a Fourier series f (n)un for which 2 f (n) = f 2 2 , and conversely. The convergence of the series is not an obvious consequence.