xii Preface Remark 0.0.1. Note that there is a tradeoff here: if one wants to keep as many useful laws of algebra as one can, then one can add in infinity, or have negative numbers, but it is difficult to have both at the same time. Because of this tradeoff, we will see two overlapping types of measure and integration theory: the non-negative theory, which involves quantities taking values in [0, +∞], and the absolutely integrable theory, which involves quantities tak- ing values in (−∞, +∞) or C. For instance, the fundamental convergence theorem for the former theory is the monotone convergence theorem (The- orem 1.4.43), while the fundamental convergence theorem for the latter is the dominated convergence theorem (Theorem 1.4.48). Both branches of the theory are important, and both will be covered in later notes. One important feature of the extended non-negative real axis is that all sums are convergent: given any sequence x1,x2,... [0, +∞], we can always form the sum n=1 xn [0, +∞] as the limit of the partial sums ∑N n=1 xn, which may be either finite or infinite. An equivalent definition of this infinite sum is as the supremum of all finite subsums: n=1 xn = sup F ⊂N,F finite n∈F xn. Motivated by this, given any collection (xα)α∈A of numbers [0, +∞] indexed by an arbitrary set A (finite or infinite, countable or uncountable), we can define the sum α∈A by the formula (0.1) α∈A = sup F ⊂A,F finite α∈F xα. Note from this definition that one can relabel the collection in an arbitrary fashion without affecting the sum more precisely, given any bijection φ : B A, one has the change of variables formula (0.2) α∈A = β∈B xφ(β). Note that when dealing with signed sums, the above rearrangement iden- tity can fail when the series is not absolutely convergent (cf. the Riemann rearrangement theorem). Exercise 0.0.1. If (xα)α∈A is a collection of numbers [0, +∞] such that α∈A ∞, show that = 0 for all but at most countably many α A, even if A itself is uncountable.
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