1. Weak Convergence of Measures: Applications in Probability

1. Introduction. Let Ω be the unit interval [0, 1], let ℬ consist of the Borel sets in [0, 1], and let P denote Lebesgue measure on ℬ, so that (Ω, ℬ, P) is a probability space. DefineXt(ω)=0for 0≦t≦1 and ω ∈ Ω, and defineYt(ω)=0ift≠ω,1ift=ωfor t and ω in the same ranges. Then P{Xt=0}=P{Yt=0}=1 , so that the stochastic processes {Xt:0≦t≦1} and {Yt:0≦t≦1} have the same finite-dimensional distributions, in the sense thatP{Xt1≦x1,⋯,Xtk≦xk}=P{Yt1≦x1,⋯,Ytk≦xk}for all choices of the ti and xj . On the other hand,sup0≦t≦1Xt(ω)=0,sup0≦t≦1Yt(ω)=1for all ω.