On a definition of random sequences with respect to conditional probability

We study a universal Martin-Löf test with respect to a computable probability on a product space. Then, we define random sequences with respect to a conditional probability by using a section of the set of random points of product space. We show that (1) our definition is consistent with Fubini’s theorem, and (2) it is equivalent to the relative notion of randomness under a condition. This is an extension of Lambalgen’s theorem (1987) to a correlated probability.