Generalization of van Lambalgen's theorem and blind randomness for conditional probability

Lambalgen’s theorem (1987) [6] says that a pair of sequences (x, y) ∈ Ω2 is Martin-Löf (ML) random w.r.t. product of uniform measures iff x is ML-random and y is ML-random relative to x, where Ω is the set of infinite binary sequences. In this paper we give a generalized form of the Lambalgen’s theorem using the notion of blind randomness. Let S be the set of finite binary strings and ∆(s) := {sx|x ∈ Ω} for s ∈ S, where sx is the concatenation of s and x. Let ∆(x, y) := ∆(x)× ∆(y) for x, y ∈ S. For a probability P on Ω, we write P (s) := P (∆(s)) for s ∈ S. For a probability P on X × Y,X = Y = Ω, let P (x, y) := P (∆(x, y)) for x, y ∈ S and PX , PY be the marginal distributions on X and Y , respectively. For x, y ∈ S, we write x ⊑ y is x is a prefix of y. Let N be the set of natural numbers. In Vovk and Vyugin (1993) [7], they generalized Lambalgen’s theorem as follows (actually they show a different form of the following theorem with parametric models, however the following form is easily derived from them)