Notation xiii We will rely frequently on the following basic fact (a special case of the Fubini-Tonelli theorem, Corollary 1.7.23): Theorem 0.0.2 (Tonelli’s theorem for series). Let (xn,m)n,m∈N be a doubly infinite sequence of extended non-negative reals xn,m ∈ [0, +∞]. Then (n,m)∈N2 xn,m = ∞ n=1 ∞ m=1 xn,m = ∞ m=1 ∞ n=1 xn,m. Informally, Tonelli’s theorem asserts that we may rearrange infinite series with impunity as long as all summands are non-negative. Proof. We shall just show the equality of the first and second expressions the equality of the first and third is proven similarly. We first show that (n,m)∈N2 xn,m ≤ ∞ n=1 ∞ m=1 xn,m. Let F be any finite subset of N2. Then F ⊂ {1,...,N} × {1,...,N} for some finite N, and thus (by the non-negativity of the xn,m) (n,m)∈F xn,m ≤ (n,m)∈{1,...,N}×{1,...,N} xn,m. The right-hand side can be rearranged as N n=1 N m=1 xn,m, which is clearly at most ∑∞ n=1 ∑∞ m=1 xn,m (again by non-negativity of xn,m). This gives (n,m)∈F xn,m ≤ ∞ n=1 ∞ m=1 xn,m, for any finite subset F of N2, and the claim then follows from (0.1). It remains to show the reverse inequality ∞ n=1 ∞ m=1 xn,m ≤ (n,m)∈N2 xn,m. It suffices to show that N n=1 ∞ m=1 xn,m ≤ (n,m)∈N2 xn,m for each finite N.
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