xiv Preface Fix N. As each ∑∞ m=1 xn,m is the limit of ∑M m=1 xn,m, the left-hand side is the limit of ∑N n=1 ∑M m=1 xn,m as M ∞. Thus it suffices to show that N n=1 M m=1 xn,m (n,m)∈N2 xn,m for each finite M. But the left-hand side is (n,m)∈{1,...,N}×{1,...,M} xn,m, and the claim follows. Remark 0.0.3. Note how important it was that the xn,m were non-negative in the above argument. In the signed case, one needs an additional assump- tion of absolute summability of xn,m on N2 before one is permitted to in- terchange sums this is Fubini’s theorem for series, which we will encounter later in this text. Without absolute summability or non-negativity hypothe- ses, the theorem can fail (consider, for instance, the case when xn,m equals +1 when n = m, −1 when n = m + 1, and 0 otherwise). Exercise 0.0.2 (Tonelli’s theorem for series over arbitrary sets). Let A, B be sets (possibly infinite or uncountable), and (xn,m)n∈A,m∈B be a doubly infinite sequence of extended non-negative reals xn,m [0, +∞] indexed by A and B. Show that (n,m)∈A×B xn,m = n∈A m∈B xn,m = m∈B n∈A xn,m. (Hint: Although not strictly necessary, you may find it convenient to first establish the fact that if n∈A xn is finite, then xn is non-zero for at most countably many n.) Next, we recall the axiom of choice, which we shall be assuming through- out the text: Axiom 0.0.4 (Axiom of choice). Let (Eα)α∈A be a family of non-empty sets Eα, indexed by an index set A. Then we can find a family (xα)α∈A of elements of Eα, indexed by the same set A. This axiom is trivial when A is a singleton set, and from mathematical induction one can also prove it without difficulty when A is finite. However, when A is infinite, one cannot deduce this axiom from the other axioms of set theory, but must explicitly add it to the list of axioms. We isolate the countable case as a particularly useful corollary (though one which is strictly weaker than the full axiom of choice): Corollary 0.0.5 (Axiom of countable choice). Let E1,E2,E3,... be a se- quence of non-empty sets. Then one can find a sequence x1,x2,... such that xn En for all n = 1, 2, 3,....
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