The support of a measure λ is the smallest closed set F such that λ(F c ) = 0. Let P be a Wiener measure on C[0, 1], i.e., the law of a Brownian motion on [0, 1]. Use Exercise 13.4 to prove that the...

The support of a measure λ is the smallest closed set F such that λ(F^c) = 0. Let P be a Wiener measure on C[0, 1], i.e., the law of a Brownian motion on [0, 1]. Use Exercise 13.4 to prove that the support of P is all of C[0, 1].

Let (S, d) be a complete separable metric space and let R be a subset of S. Then (R, d) is also a metric space. If X_n
converges weakly to X with respect to the topology of (S, d) and each X_n
and X take values in R, does X_n
converge weakly to X with respect to the topology of (R, d)? Does the answer change if R is a closed subset of S? If X_n
and X take values in R and X_n
converges weakly to X with respect to the topology of (R, d), does X_n
converge weakly to X with respect to the topology of (S, d)? What if R is a closed subset of S?

Give a proof of Theorem 32.2 using (32.7) in place of.