Tightness of Probabilities On $C(\lbrack 0, 1 \rbrack; \mathscr{Y}')$ and $D(\lbrack 0, 1 \rbrack; \mathscr{Y}')$

Abstract

Let $C_\mathscr{I}' = C(\lbrack 0, 1\rbrack; \mathscr{I}')$ be the space of all continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$, where $\mathscr{L}'$ is the topological dual of the Schwartz space $\mathscr{I}$ of all rapidly decreasing functions. Let $C$ be the Banach space of all continuous functions on $\lbrack 0, 1\rbrack$. For each $\varphi \in \mathscr{I}, \Pi_\varphi$ is defined by $\Pi_\varphi:x \in C_\mathscr{I}' \rightarrow x. (\varphi) \in C$. Given a sequence of probability measures $\{P_n\}$ on $C_\mathscr{I}'$ such that for each $\varphi \in \mathscr{I}, \{P_n\Pi^{-1}_\varphi\}$ is tight in $C$, we prove that $\{P_n\}$ itself is tight in $C_\mathscr{I}'.$ A similar result is proved for the space of all right continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$.