44 1. Measure theory Definition 1.3.5 (Almost everywhere and support). A property P (x) of a point x ∈ Rd is said to hold (Lebesgue) almost everywhere in Rd, or for (Lebesgue) almost every point x ∈ Rd, if the set of x ∈ Rd for which P (x) fails has Lebesgue measure zero (i.e. P is true outside of a null set). We usually omit the prefix Lebesgue, and often abbreviate “almost everywhere” or “almost every” as a.e. Two functions f, g : Rd → Z into an arbitrary range Z are said to agree almost everywhere if one has f(x) = g(x) for almost every x ∈ Rd. The support of a function f : Rd → C or f : Rd → [0, +∞] is defined to be the set {x ∈ Rd : f(x) = 0} where f is non-zero. Note that if P (x) holds for almost every x, and P (x) implies Q(x), then Q(x) holds for almost every x. Also, if P1(x),P2(x),... are an at most countable family of properties, each of which individually holds for almost every x, then they will simultaneously be true for almost every x, because the countable union of null sets is still a null set. Because of these properties, one can (as a rule of thumb) treat the almost universal quantifier “for almost every” as if it was the truly universal quantifier “for every”, as long as one is only concatenating at most countably many properties together, and as long as one never specialises the free variable x to a null set. Observe also that the property of agreeing almost everywhere is an equivalence relation, which we will refer to as almost everywhere equivalence. In An epsilon of room, Vol. I, we will also see the notion of the closed support of a function f : Rd → C, defined as the closure of the support. The following properties of the simple unsigned integral are easily ob- tained from the definitions: Exercise 1.3.1 (Basic properties of the simple unsigned integral). Let f, g : Rd → [0, +∞] be simple unsigned functions. (i) (Unsigned linearity) We have Simp Rd f(x) + g(x) dx = Simp Rd f(x) dx + Simp Rd g(x) dx and Simp Rd cf(x) dx = c × Simp Rd f(x) dx for all c ∈ [0, +∞]. (ii) (Finiteness) We have Simp Rd f(x) dx ∞ if and only if f is finite almost everywhere, and its support has finite measure.

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