1.7. Outer measures, pre-measures, product measures 149 to both discrete and continuous random processes, even if they have infinite length. The most important result about product measure, beyond the fact that it exists, is that one can use it to evaluate iterated integrals, and to inter- change their order, provided that the integrand is either unsigned or abso- lutely integrable. This fact is known as the Fubini-Tonelli theorem, and is an absolutely indispensable tool for computing integrals, and for deducing higher-dimensional results from lower-dimensional ones. In this section we will, however, omit a very important way to construct measures, namely the Riesz representation theorem, which is discussed in §1.10 of An epsilon of room, Vol. I. 1.7.1. Outer measures and the Carath´ eodory extension theorem. We begin with the abstract concept of an outer measure. Definition 1.7.1 (Abstract outer measure). Let X be a set. An abstract outer measure (or outer measure for short) is a map μ∗ : 2X → [0, +∞] that assigns an unsigned extended real number μ∗(E) ∈ [0, +∞] to every set E ⊂ X which obeys the following axioms: (i) (Empty set) μ∗(∅) = 0. (ii) (Monotonicity) If E ⊂ F , then μ∗(E) ≤ μ∗(F ). (iii) (Countable subadditivity) If E1,E2,... ⊂ X is a countable se- quence of subsets of X, then μ∗( ∞ n=1 En) ≤ ∑∞ n=1 μ∗(En). Outer measures are also known as exterior measures. Thus, for instance, Lebesgue outer measure m∗ is an outer measure (see Exercise 1.2.3). On the other hand, Jordan outer measure m∗,(J) is only finitely subadditive rather than countably subadditive and thus is not, strictly speaking, an outer measure for this reason this concept is often referred to as Jordan outer content rather than Jordan outer measure. Note that outer measures are weaker than measures in that they are merely countably subadditive, rather than countably additive. On the other hand, they are able to measure all subsets of X, whereas measures can only measure a σ-algebra of measurable sets. In Definition 1.2.2, we used Lebesgue outer measure together with the notion of an open set to define the concept of Lebesgue measurability. This definition is not available in our more abstract setting, as we do not necessar- ily have the notion of an open set. An alternative definition of measurability was put forth in Exercise 1.2.17, but this still required the notion of a box or an elementary set, which is still not available in this setting. Nevertheless, we can modify that definition to give an abstract definition of measurability:

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