OAI.IndependentProducts.main_general

The theorem states that, for any Borel measure μ on ℝ satisfying the multiplier hypotheses, the closed real L¹ span of the path products is infinite-dimensional, and no equivalent norm on it is AUC. The hypotheses say that μ is a probability measure, almost every value is positive, μ is not almost surely equal to any constant, its mean is 1, and the identity function has a finite second moment (is in L²). Vertices are finite lists of positive integers, a sample assigns a real number to each nonroot vertex, and the product measure is the infinite product of independent copies of μ, one per nonroot vertex. The path product of a vertex v multiplies the sample values at the nonempty prefixes of v, and is 1 for the root. productSpan(μ) is the topological closure, in L¹ of the product measure, of the real linear span of the classes equal almost everywhere to some path product. The theorem concludes first that productSpan(μ) is not finite-dimensional over ℝ. Second, for every seminorm N on productSpan(μ) that is an equivalent norm, meaning there are constants 0<a≤b with a‖x‖≤N(x)≤b‖x‖ for all x, N fails the AUC property, which requires that, for every t>0, the one-sided asymptotic modulus of N at t is strictly positive. That modulus is the infimum over N-unit vectors x of the supremum over closed finite-codimension subspaces F of the infimum of N(x+ty)−1 over y in F with N(y)=1. The statement is given as an admitted theorem without proof.