6 2. PRELIMINARIES For a compact space K, M pKq is the Banach space of all signed Borel measures on K. By the Riesz representation theorem, the dual CpKq˚ can be identified with the subspace MrpKq of M pKq consisting of regular measures see e.g. [12, Theorem 7.3.5], [22, Theorem IV.6.3] or [13, Theorem III.5.7]. (If e.g. K is compact and metrizable, then every signed measure is regular, so CpKq˚ M pKq, see [12, Propositions 7.1.12, 7.2.3, 7.3.3].) Dr0, 1s denotes the linear space of functions r0, 1s Ñ R that are right-continuous with left limits, see e.g. [6, Chapter 3]. The norm is supr0,1s |f |. See further Sec- tion 9.1. δs denotes the Dirac measure at s, as well as the corresponding point evaluation f Þ Ñ f psq seen as a linear functional on a suitable function space. pΩ, F, Pq denotes the underlying probability space where our random variables are defined ω denotes an element of Ω. We assume that pΩ, F, Pq is complete. We let ¯ E denote the upper integral of a, possibly non-measurable, real-valued function on Ω: (2.1) ¯Y E :“ inf E Z : Z ě Y and Z is measurable ( . (If Y is measurable, then ¯Y E E Y .) In particular, if X is B-valued, then ¯}X} E ă 8 if and only if there exists a positive random variable Z with }X} ď Z and E Z ă 8. The exponent k is, unless otherwise stated, an arbitrary fixed integer ě 1, but the case k 1 is often trivial. For applications, k 2 is the most important, and the reader is adviced to primarily think of that case. 2.2. Measurability A B-valued random variable is a function X : Ω Ñ B defined on some prob- ability space pΩ, F, Pq. (As said above, we assume that the probability space is complete.) We further want X to be measurable, and there are several possibilities to consider we will use the following definitions. Definition 2.1. Let X : Ω Ñ B be a function on some probability space pΩ, F, Pq with values in a Banach space B. (i) X is Borel measurable if X is measurable with respect to the Borel σ-field B on B, i.e., the σ-field generated by the open sets. (ii) X is weakly measurable if X is measurable with respect to the σ-field Bw on B generated by the continuous linear functionals, i.e., if xx˚, Xy is measurable for every P B˚. (iii) X is a.s. separably valued if there exists a separable subspace B1 Ď B such that X P B1 a.s. (iv) X is weakly a.s. separably valued if there exists a separable subspace B1 Ď B such that if P and K B1, then x˚pXq 0 a.s. (v) X is Bochner measurable if X is Borel measurable and a.s. separably valued. Remark 2.2. X is Bochner measurable if and only if X is Borel measurable and tight, i.e., for every ε ą 0, there exists a compact subset K Ă B such that PpX P Kq ą 1 ´ ε, see [6, Theorem 1.3]. (This is also called Radon.) Some authors use the name strongly measurable. Moreover, X is Bochner measurable if and only if there exists a sequence Xn of measurable simple functions Ω Ñ B such
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