By Roger Godement
Research quantity IV introduces the reader to practical research (integration, Hilbert areas, harmonic research in workforce conception) and to the equipment of the speculation of modular services (theta and L sequence, elliptic capabilities, use of the Lie algebra of SL2). As in volumes I to III, the inimitable variety of the writer is recognizable right here too, not just due to his refusal to put in writing within the compact sort used these days in lots of textbooks. the 1st half (Integration), a sensible mixture of arithmetic stated to be 'modern' and 'classical', is universally invaluable while the second one half leads the reader in the direction of a truly lively and really good box of study, with probably huge generalizations.
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Additional resources for Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext)
Bn = An − An ∩ (A1 ∪ . . ∪ An−1 ) , we get a partition of P into Borel sets of diameter ≤ 2r. Since f is measurable, the sets En = f −1 (Bn ) form a partition of X into measurable sets on which the function f is constant up to 2r. For each n, we choose some xn ∈ En and we define a step map f from X to P by the condition g(x) = f (xn ) on En . Then the uniform distance d(f, g) = sup d[f (x), g(x)] is clearly ≤ 2r. Giving values of type 1/n to r, we thus see that f is the uniform limit in X of a sequence of measurable step functions gn with values in P .
In this case: Theorem 9 (dominated convergence12 ). Let (fn ) be a sequence of functions inLp converging ae. to a function f . Suppose that there is a function F ≥ 0 such that Np (F ) < +∞ |fn (x)| ≤ F (x) ae. for all n . & Then f is in Lp and lim Np (f − fn ) = 0 , lim µ(fn ) = µ(f ) if p = 1 . It suffices to show that (fn ) is a Cauchy sequence with respect to convergence in mean. , lemma 4 shows that the functions gn (x) = sup |fi (x) − fj (x)| i,j≥n are in Lp , and lemma 2 that this decreasing sequence of positive functions converges in mean in Lp .
However, |fi (x) − fj (x)| ≤ gn and so Np (fi − fj ) ≤ Np (gn ) for i, j ≥ n . This gives Cauchy’s criteria and convergence in Lp , qed. Compare with theorem 19 of Chap. V, n◦ 23. Corollary. Let p and q be two real numbers such that 1 ≤ p, q < +∞ . For all functions f ∈ Lp ∩ Lq , there exists a sequence of functions fn ∈ L(X) converging to f almost everywhere, as well as in Lp and in Lq . Indeed, it is possible to extract from L(X) two sequences (gn ) and (hn ) that are convergent to f ae. and such that f is the limit in Lp (resp.