By Tao T.

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Nevertheless, there is a non-trivial version of the above theory that can be applied to finite sets (cf. 3 of Structure and Randomness). 9 (Finitary analogue of the Lebesgue-Radon-Nikodym theorem). Let Xn be a sequence of finite sets (and with the discrete σ-algebra), and for each n, let mn be the uniform distribution on Xn , and let µn be another probability measure on Xn . 12) µn = µn,ac + µn,sc + µn,pp where (i) (Uniform absolute continuity) For every ε > 0, there exists δ > 0 (independent of n) such that µn,ac (E) ≤ ε whenever mn (E) ≤ δ, for all n and all E ⊂ Xn .

Lp is a Banach space for every 1 ≤ p ≤ ∞. ∞ Proof. 7, it suffices to show that any series n=1 fn of functions in Lp which is absolutely convergent, is also conditionally convergent. This is easy in the case p = ∞ and is left as an exercise. ∞ p In the case 1 ≤ p < ∞, we write M := n=1 fn L , which is a finite quantity by hypothesis. By the triangle inequality, we have N n=1 |fn | Lp ≤ M for all N . 21), we conclude n=1 |fn | L ≤ M . In particular, n=1 fn (x) is absolutely convergent for almost every x.

In order to motivate the discussion of the more general classes of spaces, we will first focus on the most special class - that of (real and complex) Hilbert spaces. These spaces can be viewed as generalisations of (real and complex) Euclidean spaces such as Rn and Cn to infinite-dimensional settings, and indeed much of one’s Euclidean geometry intuition concerning lengths, angles, orthogonality, subspaces, etc. will transfer readily to arbitrary Hilbert spaces; in contrast, this intuition is not always accurate in the more general vector spaces mentioned above.