Conditional Expectations of Random Variables Without Expectations

Let (Ω, F, P) be a probability space, and let X be a random variable defined on (Ω, F, P). If A is a sub σ-field of F, then E(X A) is the a.s. unique A measurable function such that, for all A ε A, {equation*}{1}∫_A X dP = ∫_A E(X A) dP,{equation*} provided $EX$ is defined. ([2], p. 341). If $EX$ is not defined, that is, if EX⁺ = EX⁻ = ∞, we may then define E(X A) = E(X⁺ A) - E(X⁻ A), provided the difference is defined almost surely ([2], p. 342). We show that this is the only reasonable definition of E(X A) (Lemma 2), and exhibit several apparent pathologies, akin to the fact that a conditionally convergent series of real numbers may be re-ordered to give any sum. If X is any random variable with a continuous distribution such that $EX$ is not defined, then we can find an A ⊂ F such that E(X A) = 0 a.s. (Theorem 1), and if Y is any random variable independent of X, we can find an A ⊂ F such that E(X A) = Y a.s. (Theorem 2). In fact, if X₁, X₂, ⋯ is a sequence of independent random variables such that for n 2, EXₙ is not defined and Xₙ has a continuous distribution, we can find a sequence of σ-fields A₁ ⊂ A₂ ⊂ ⋯ ⊂ F such that X₁, ⋯, Xₙ are Aₙ measurable and E(Xn + 1 Aₙ) = Xₙ a.s. (Theorem 3). The sequence ₙ, Fₙ, n = 1, 2, ⋯\ is not a martingale however, since for m > n + 1, E(Xₘ Fₙ) is not defined. We remark that the standard theorem on iterated conditional expectations, ([2], p. 350) which says that if A ⊂ B ⊂ F then E(X A) = E(E(X B) A) a.s. is valid only if E(X A) is defined a.s.