0-1 Laws of a Probability Measure on a Locally Convex Space

We introduce several 0-1 laws for a cylindrical probability measure /j on a locally convex Hausdorff space E and examine the equivalence of them. We show that the following 0-1 laws are equivalent: (a) for every x'n in E' , [t(x; (x'n(x)) GCO) =0 or 1, (b) for every x'n in Ef , fjt(x; (x'n(x)) eq) =0 or 1, and (c) for every x'n in E' . j"(*; (^M) G/oo)=0 or 1. We also show that each of (a), (b) and (c) implies: (d) for every x'n in E', fj(x; (x'n (x) ) e lp) = 0 or 1. If ^ is a Radon probability measure, then (a), (b) and (c) are equivalent to: (e) for every lower semi-continuous semi-norm N, }=Q or 1. § 1. 0-1 Laws In this section, we present several 0-1 laws which appear in the probability