# Natural & important probability measures on $\mathcal{C}[0,1]$, in particular the Wiener measure

Which probability measures on $\mathbf{\mathcal{C}[0,1]}$ are known? (Here $\mathcal{C}[0,1]$ is the space of continuous real-valued functions defined on the unit interval.)

I'm pretty sure the Wiener measure (or maybe it's "a" Wiener measure) is one such measure whose random elements are Brownian motion paths (or something like that, please correct me if this, or anything else I've said, is incorrect). Where can I read about the Wiener measure as a measure (instead of just a process with an implicit measure; I want explicit analysis of the measure)? Planet math has a little bit about this, but I'd like more detail.

To answer my "where can I read..." question, feel free to just give an explanation yourself, if you're so inclined.

Edit. At the behest of Nate (see the comments) I'll try to be more explicit. In Cantor space, $\{0,1\}^{\mathbb{N}}$, the uniform (i.e. Lebesgue) measure is the most natural. Is there a most natural measure in $\mathbf{\mathcal{C}[0,1]}$? After that, the Bernoulli measures are most natural measures on Cantor space. So I'd like some examples of "natural" measures on $\mathcal{C}[0,1]$. As Nate said in his comment, the space of measures is the same as the space of continuous processes. Since I'm not so "process-inclined", I'd prefer answers and references to be in the language of measures rather than processes. However, if there's a process that is very natural (or at least important), but who's corresponding measure is harder to work with and therefore unavailable in the literature, a reference to or description of that process will still be appreciated.

Nate's comment also led to this question, an answer to which might render void my request for answers to be in the language of measures:
What exactly is the correspondence between measures and continuous stochastic processes?

• I should note that I'm aware of Dirac measures and convex combinations thereof. So unless you think there's something about them that I might not know, no need to remind me of them. – Quinn Culver May 7 '12 at 23:39
• Billingsley, "Convergence of Probability Measures" should cover what you are interested in. – Michael Greinecker May 7 '12 at 23:44
• @MichaelGreinecker Yes, I think you're right, at least with respect to Winer measure. What about other (nontrivial) measures, though? – Quinn Culver May 8 '12 at 0:04
• Probability measures on $C([0,1])$ are in one-to-one correspondence with continuous stochastic processes $\{X_t : 0 \le t \le 1\}$, so in some sense there are no other examples. Statements about processes can be rephrased as statements about measures and vice versa, though sometimes one seems more natural than the other. It would help if you could you be more specific about what you are looking for. – Nate Eldredge May 8 '12 at 0:08
• @NateEldredge Thanks. I've edited my question to show better what I'm after. Please let me know if you think more detail is still necessary. – Quinn Culver May 8 '12 at 0:22

Regarding the identification between processes and measures:

Suppose $$\{X_t : 0 \le t \le 1\}$$ is a continuous stochastic process on a probability space $$(\Omega, \mathcal{F}, P)$$. Then there is a natural map $$X : \Omega \to C([0,1])$$ defined by $$X(\omega)(t) = X_t(\omega)$$. I claim this map is measurable with respect to the Borel $$\sigma$$-algebra on $$C([0,1])$$. Let $$B = B(x,r)$$ be an open ball in $$C([0,1])$$. Then $$X^{-1}(B)$$ is the set of all $$\omega$$ such that $$|X_t(\omega) - x(t)| for all $$t \in [0,1]$$. By continuity, it is sufficient to check this holds for all rational $$t$$, and so we can write $$X^{-1}(B) = \bigcap_{t \in [0,1] \cap \mathbb{Q}} X_t^{-1}((x(t)-r, x(t)+r))$$ which is measurable in $$\Omega$$. Since the open balls generate the Borel $$\sigma$$-algebra, $$X$$ is measurable. Now let $$\mu = P \circ X^{-1}$$ be the pushforward of $$P$$ onto $$C([0,1])$$. $$\mu$$ is called the law or distribution of the process $$\{X_t\}$$.

Conversely, if $$\mu$$ is a Borel probability measure on $$C([0,1])$$, then $$(C([0,1]), \mathcal{B}, \mu)$$ is a probability space. For $$t \in [0,1]$$, define $$X_t : C([0,1]) \to \mathbb{R}$$ by $$X_t(x) = x(t)$$. Each $$X_t$$ is measurable (indeed a continuous linear functional) so $$\{X_t : 0 \le t \le 1\}$$ is a stochastic process, which is clearly continuous in $$t$$. It is also easy to check that the law of this $$\{X_t\}$$ is in fact $$\mu$$, since the map $$X$$ constructed above is just the identity in this case.

There isn't a measure on $$C([0,1])$$ which is quite as natural as Lebesgue measure on $$\{0,1\}^\mathbb{N}$$. In particular, there is no translation-invariant Borel probability measure (nor even one which is $$\sigma$$-finite). However, Wiener measure is a pretty important example, since it is "universal" in many ways. For instance, Donsker's invariance principle says it arises as the scaling limit of measures corresponding to random walks (with, say, piecewise linear interpolation). Karatzas and Shreve, Brownian Motion and Stochastic Calculus has the full statement (note that the Wikipedia page on Donsker's theorem describes a different result).

Wiener measure is an example of a Gaussian measure, of which there are many. I'll put in a plug for these lecture notes of mine where there is some more information on Gaussian measures.

Whenever $K(s,t)$ is a positive-definite function (for example, the Fourier transform of a positive measure, by Bochner's theorem), you can define an associated Gaussian process on $\mathcal{C}[0,1]$ with covariance function $K(s,t)$.

While the Wiener process is probably the best-known Gaussian process (to mathematicians, at least), there are many other examples. GPs are used quite extensively in machine learning.