[ m at h . FA ] 6 D ec 1 99 9 Tangent Sequences in Orlicz and Rearrangement Invariant Spaces

Let (Fn) be an increasing sequence of σ-algebras on some probability space (Ω,F , P ). We will assume that F0 = {∅,Ω}. A sequence (fn) of random variables is called (Fn)adapted if fn is Fn-measurable for each n ≥ 1. In the sequel we will simply write ‘adapted’ if there is no risk of confusion. For any sequence (fn) of random variables, we will write f = supn |fn| and f ∗ n = max1≤k≤n |fk|. Throughout the paper all equalities or inequalities between random variables are assumed to hold almost surely. Given a σ-algebra A ⊂ F and an integrable random variable f , we will denote the conditional expectation of f given A by EAf . If A = Fk then we will simply write Ekf for EFkf . The conditional distribution of a random variable f given A is denoted by L(f ∣