The essential equivalence of pairwise and mutual conditional independence

For a large collection of random variables, pairwise conditional independence and mutual conditional independence are shown to be essentially equivalent — i.e., equivalent to up to null sets. Unlike in the finite setting, a large collection of random variables remains essentially conditionally independent under further conditioning. The essential equivalence of pairwise and multiple versions of exchangeability also follows as a corollary. Our result relies on an iterated extension of Bledsoe and Morse's completion of the product of two measure spaces. Part of this work was done while Peter Hammond was visiting the National University of Singapore in March–April 2004. The final version was completed while Yeneng Sun was on sabbatical leave at the University of Illinois at Urbana–Champaign and Stanford University in October 2004 – May 2005.