# Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in $C[a,b]$.

Are singleton sets like $\{f\}$ in $C[a,b]$ measurable with respect to $\mathcal{A}$ and are they null sets?

• Got the part with measurability. How can I show it is indeed a null set? I am trying to cover $\{f\}$ in a cylinder set -- since I can calculate the Wiener measure of such sets -- of size $\epsilon$ and let the $\epsilon$ tend to zero, but I have not been able to finish it off. – basicHTML Aug 4 '15 at 14:04
• @basicHTML There is a similar approach in terms of cylindric sets: if you have any given $f$ and $t>0$, then $\{ f \}$ is contained in $C_\delta = \{ g : g(t) \in [f(t)-\delta,f(t)+\delta] \}$ for every $\delta > 0$. But these are cylindric sets of decaying measure. – Ian Aug 4 '15 at 14:37