Conditional expectations and submartingale sequences of random Schwartz distributions

Let (Ω, Σ, P) be a fixed complete probability space, $\text{D}$ the real Schwartz space, and $\text{D′}$ its strong dual. $\text{D}$ and $\text{D′}$ are partially ordered by $\text{C}$ and $\text{C′}$ respectively, where $\text{C}$ is the positive cone of nonnegative functions in $\text{D}$ and $\text{C′}$ its dual in $\text{D′}$. $\text{C}$ is a strict $\text{B}$-cone and $\text{C′}$ is normal, where $\text{B}$ is the family of all bounded subsets of $\text{D}$. If X, Y are two random Schwartz distributions, then X ≤ Y if and only if Y(ω) ? X(ω) ∈ $\text{D′}$ for almost all $\text{ω ∈ Ω(P)}$. Integrability of random Schwartz distributions and properties of such integrals are discussed. The monotone convergence theorem, the dominated convergence theorem, and Fatou's lemma are proved. The existence of conditional expectations of integrable random Schwartz distributions relative to a given sub σ-field of Σ is shown. Properties of conditional expectations are discussed and the conditional form of the monotone convergence theorem is proved. Sub(super)-martingale sequences are defined via the partial order relations introduced above, and a convergence theorem is given. The notion of a potential is introduced and the Riesz decomposition theorem is proved.