Get Analysis IV: Integration and Spectral Theory, Harmonic PDF

By Roger Godement

ISBN-10: 3319169076

ISBN-13: 9783319169071

Research quantity IV introduces the reader to useful research (integration, Hilbert areas, harmonic research in workforce idea) and to the tools 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 due to his refusal to write down within the compact variety used these days in lots of textbooks. the 1st half (Integration), a smart mixture of arithmetic acknowledged to be 'modern' and 'classical', is universally worthwhile while the second one half leads the reader in the direction of a really lively and really good box of analysis, with in all likelihood large generalizations.

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Extra resources for Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext)

Example text

If, moreover, the sequence fn (x) converges almost everywhere to a limit g(x), then f (x) = g(x) almost everywhere. § 2. Lp Spaces 25 Being a closed subspace of a complete space, Lp is complete. The second proposition of the statement is here too theorem 5 in the special case of Lp . The appearance of “ exceptional ” null sets in these statements cannot be avoided: the behaviour everywhere of a sequence or of a series cannot be expected to follow from integral computations. Similarly, statistics provides no information on an individual case (the reason why Condorcet had already realized around 1750 that statistics are only useful for political leaders).

2. Lp Spaces 29 Apply the previous result to the functions inf(f1 , . . , fn ); we already know (lemma 1) they are in Lp . Lemma 4. The upper envelope of a countable family of positive functions fn ∈ Lp is in Lp if and only if there is function F ≥ 0 such that11 Np (F ) < +∞ fn (x) ≤ F (x) ae. & for all n . The condition is necessary: for F take the upper envelope of the functions fn . If it holds, the functions gn = sup(f1 , . . , whence Np (gn ) ≤ Np (F ) < +∞. Apply theorem 8 to functions gn completes the proof.

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 .

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Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext) by Roger Godement

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