Suppose P and P are measures supported on D[0, 1] that agree on all cylindrical subsets of D[0, 1]. In other words, all the finite-dimensional distributions agree. Prove that P = P on D[0, 1]. Show...

Suppose P and P are measures supported on D[0, 1] that agree on all cylindrical subsets of D[0, 1]. In other words, all the finite-dimensional distributions agree. Prove that P = P on D[0, 1].

Show that the following are continuous functions on the space D[0, 1].

Let P be a Poisson process with parameter λ. Prove that

converges weakly with respect to the topology of D[0, 1] as n → ∞ to a Brownian motion.