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Let $\Omega$ be the space of continuous paths from $\mathbb{R}$ to $\mathbb{R}^n$. By a famous result, it is known that $\Omega$ is a measure space if we equip it with the "Wiener measure" (see this link).

In a descriptive or intuitive sense, what does the Wiener measure tell us about a given path $\alpha$?

It seems that if $\alpha$ covers a 'lot of distance', then it should have larger measure than a path that covers 'a shorter distance' (though if $\alpha$ is bad enough, then maybe the idea of distance is not so meaningful). In particular, does the Wiener measure generalize the notion of arc-length?

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Not at all. In fact, the Wiener measure of a singleton is just zero. The Wiener measure of a set is just the probability that a Wiener process trajectory is a member of that set. Thus for instance the set of functions which are differentiable at some point in $\mathbb{R}_+$ has Wiener measure zero. Similarly the set of monotone functions has Wiener measure zero.

Some simple examples of sets with Wiener measure in $(0,1)$ are the cylindric sets: these are the sets of the form $\{ f : f(t_1) \in I_1,f(t_2) \in I_2,\dots,f(t_n) \in I_n \}$ where $t_i \geq 0$ and $I_i$ are intervals. In fact these sets generate the Borel $\sigma$-algebra on the continuous functions, so the Wiener measure of any set of continuous functions can be realized as a limit using cylindric sets.

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  • $\begingroup$ I'm confused about the statement 'the Wiener measure of a set is the probability that a Wiener process trajectory is a member of that set.' If this were always true, then the measure would only take values on 0 and 1, right? $\endgroup$ – P. Turner Sep 28 '15 at 2:21
  • $\begingroup$ Also, the Wiener measure is indeed a probability measure? $\endgroup$ – P. Turner Sep 28 '15 at 2:21
  • $\begingroup$ @P.Turner No, because the Wiener process trajectory is randomly selected. It is not really any different from a random variable randomly selecting a number; instead you are just randomly selecting a continuous function. And yes, the Wiener measure is a probability measure. $\endgroup$ – Ian Sep 28 '15 at 2:40
  • $\begingroup$ Ok, cool. So if I understand correctly, if I have a set of continuous functions $S$, then the Wiener measure of $S$ is, intuitively speaking, the probability of drawing a Wiener process trajectory from $S$? $\endgroup$ – P. Turner Sep 28 '15 at 18:18
  • $\begingroup$ @P.Turner That's right. $\endgroup$ – Ian Sep 29 '15 at 0:19
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As Ian explained above, the Wiener measure of a path is indeed $0$. In general, whenever you have uncountable number of elements in your space, your probability measure can be only non-zero on a countable number of points. The other alternative would lead to the probability measure of the whole space being infinite. And in the spirit of Brownian measure, there is really no reason why you would want a probability measure which is non-zero on a countably many paths (a tiny proportion of the set of all continuous paths) and 0 on all the rest. This being said, I always find it fascinating and a bit odd that the fact of having measure $0$ for singletons in a non-discrete probability space is very well aligned with our humanly limitations:

1- Indeed when you try to measure the location of a particle in a liquid, the best you can do is to do it at discrete time steps and the location of the particle will not be exact, your machine will only be able to tell you an approximate location. This kind of measurement exactly corresponds to the cylinders mentioned by Ian which are of the form $C(T,U_I)=C(t_1,...,t_n,U_1,...,U_n)=\{\omega \in \Omega | \omega(t_i) \in U_i\}$. And Wiener measure actually exactly tells us measures of such set: $$ p(C(T,U_I))= \int_{U_1}dx_1...\int_{U_n}dx_n p(x_1,t_1)p(x_2-x_1,t_2-t_1)... $$ where $p(x-y,t)$ is the Gaussian distribution with variance $t$ and mean $y$ If you restrict to a single time $t$ and a single subset $U$, i.e the set $C(t,U)$, this is the set of all continuous paths that are in $U$ at time $t$. You can test the measure above using your intuition. Lets assume we are in 1D Brownian motion for simplicity and let $U$ be a fixed subset such that $x \in U, |x|>\delta$. Then $$ p(C(t,U))=\int_U \frac{e^{\frac{-x^2}{2t}}}{\sqrt{2 \pi t}}< vol(U) \frac{e^{\frac{-\delta^2}{2t}}}{\sqrt{2 \pi t}} $$ As $t \rightarrow 0$ then this probability goes to $0$ (exponential decays faster than $\frac{1}{t}$ blows up). Indeed if $U$ is a set far from origin, then if you dont give your particle enough time t, it will not reach there. You can test other stuff like what happens when $U$ is fixed and $t$ goes to infinity etc etc (first try to answer based on your intuition and then use the measure above, it is quite instructive). As for the measure of the whole space, $p(\Omega)=\frac{1}{2}(erf(\infty)-erf(-\infty))=1$.

2- This situation is particular to all the cases where we have continuous observables. If you try to formulate a dart throwing person as a probabilistic system, the observables you get will be locations on a Disk (or some copies of it). You will again see that the probability of getting a dart at an exact point on the disk will be zero and you will have to again enquire things like what is the probability of the dart falling on this piece of location.

3- On the other hand if your observable are discrete and finite, like the number you get from a die throw, the probability you construct will end up giving non-zero weights to singletons (even actually in the countably infinite case). And then this corresponds to the fact that in such experiments we can actually measure the outcome exactly, there is no uncertainty.

So from a philosophical point of view, we humans are really discrete and finite creatures. However idealised forms of what we see in nature (like turning discrete set of approximate location measurements to continuous curves or other discrete observations to non-discrete observations spaces) have more structure on them and you have more tools available to study them, hence the idealisation in mathematics. This weird correspondence between finiteness of our human nature and the fact that what you would call a "measurement" in probability theory reflecting this finiteness is something that puzzles me though.

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