By Roger Godement
Research quantity IV introduces the reader to useful research (integration, Hilbert areas, harmonic research in team conception) and to the equipment of the speculation of modular services (theta and L sequence, elliptic features, use of the Lie algebra of SL2). As in volumes I to III, the inimitable form of the writer is recognizable the following too, not just as a result of his refusal to write down within the compact sort used these days in lots of textbooks. the 1st half (Integration), a smart mix of arithmetic acknowledged to be 'modern' and 'classical', is universally priceless while the second one half leads the reader in the direction of a truly lively and really good box of analysis, with in all likelihood vast generalizations.
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Extra resources for Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext)
In some very particular cases, classical analysis can reinforce integration theory. If for example X is an open subset in C, and if µ is the usual measure dxdy, then a sequence of holomorphic functions fn ∈ Lp (X, µ) converging in Lp converges uniformly in every compact set to a holomorphic function [Chap. VIII, n◦ 4, (iv)], which is necessarily their limit in Lp . This proves that in the space Lp considered, the set of holomorphic functions is a possibly trivial closed vector subspace. But it was obviously not in order to integrate such harmless functions that Lebesgue and his successors worked for half a century.
Measurable Sets and Functions 43 Every continuous map is obviously measurable. The arguments that have led to relation (2) then prove the next result: Theorem 12. If a sequence of measurable functions with values in a metrizable and separable space converges almost everywhere, then the limit function is measurable. An immediate corollary: Theorem 13. All functions belonging to a Lp space are measurable. Because it is the limit almost everywhere of a sequence of continuous functions with values in C, the archetype of a metrizable and separable space.
Indeed, let K be a compact set. K is the disjoint union of integrable sets K ∩ En . Hence, for > 0 and all n, there is a compact set Kn ⊂ K ∩ En such that µ (K ∩ En − Kn ) < n = /2n . Since f is constant on each En , it is continuous on each Kn , hence also on Kn = K1 ∪ . . ∪ Kn for all n since the Ki are closed and pairwise disjoint. But µ (K − Kn ) = µ (K ∩ Ep − Kp ) + p≤n µ (K ∩ Ep ) . p>n The second sum is arbitrarily small for large n since µ (K ∩ Ep ) = µ(K) < +∞ p≥1 and the first one is bounded above by 1 + .