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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?

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This is easy to understand if you abstract a little further: in a Borel measure space, closed sets are measurable. In a metrizable topological space, singletons are closed. Thus if you have a Borel measure space defined over a metrizable topological space, singletons are measurable. (More generally, if you have a Borel measure space defined over a topological space in which singletons are closed, then singletons are measurable.)

And yes, singletons are null sets with respect to Wiener measure.

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  • $\begingroup$ 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. $\endgroup$ – basicHTML Aug 4 '15 at 14:04
  • $\begingroup$ @basicHTML One way would be to say something much more general: at any nonzero time, the probability that a Wiener process takes on any particular value is zero. That comes directly from normal variables being continuous. $\endgroup$ – Ian Aug 4 '15 at 14:06
  • $\begingroup$ @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. $\endgroup$ – Ian Aug 4 '15 at 14:37

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