By Roger Godement
Research quantity IV introduces the reader to sensible research (integration, Hilbert areas, harmonic research in staff thought) and to the equipment of the speculation of modular services (theta and L sequence, elliptic services, use of the Lie algebra of SL2). As in volumes I to III, the inimitable kind of the writer is recognizable the following too, not just as a result of his refusal to put in writing within the compact variety used these days in lots of textbooks. the 1st half (Integration), a smart mix of arithmetic stated 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 expert box of analysis, with in all probability huge generalizations.
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Additional resources for Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext)
Etc. Exercise. Define the measurable functions with values in the Riemann sphere C [Chap. VIII, n◦ 5, (vi)] and show that if f is measurable, then so is 1/f . Define the map x → 1/x from [−∞, +∞] to itself by agreeing, either that 1/0 = +∞, or −∞, or any other value. Show that if f : X −→ [−∞, +∞] is measurable, then so is 1/f in all three cases. 8 – Measurability and Continuity (i) Egorov’s and Lusin’s theorems. We saw above that any limit ae. of measurable functions is measurable. There is actually a more precise result whose proof, like the calculations of n◦ 7, (i), is a set theory exercise: Theorem 16 (Egorov).
This result holds for any continuous bilinear map. For example, if f and g take values in a separable Hilbert space, the function x → (f (x)|g(x)) is measurable. If f with values in a Banach space F is measurable, so is the function x → f (x) since u → u is a continuous map from F to R. If f and g, with values in a separable and metrizable space P , are measurable, so is the function x → d [f (x), g(x)] with respect to any distance compatible with the topology of P , since the map (u, v) → d(u, v) from P × P to R is continuous.
Gk = f − fnk+1 ae. So lim Np f − fnk+1 = 0 since the series gk converges in mean to g. This leads to a subsequence converging to f both in F p and almost everywhere, qed. Exercise. Let (an ) be a sequence of real numbers (for example the sequence of rational numbers). Show that the series 1/|n2 x−an |1/2 converges almost everywhere in R. The previous theorem shows that F p is a Banach space (Appendix to Chap. III, n◦ 5). In practice, it is not very useful because the functions used to define it are far too general for other non-trivial properties apart from the previous theorems to be proved.