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 ===== Final Essay: Measure on Brownian Motion =====
+We start with definition of Brownian motion.
+==== Definition (Brownian Motion) ====
+Let $\omega : [0, \infty) \rightarrow \mathbb{R}^{n}$ be a path. Given $t_1 < t_2$ and $\omega(t_1) = x_1$, the probability density for $\omega(t_2)$ is\\
+$$p(t_2-t_1, x-x_1) = \frac{1}{[4\pi (t_2-t_1) ]^{\frac{n}{2}}} e^{\frac{-|x-x_1|^2}{4(t_2-t_1)}}$$
+In addition, for any $t_1\leq t \leq t_2$, $\omega(t)$ does not depend on the trajectory of the path before $t_1$.
+The definition of Brownian motion provides a probability integral that motivates the construction of the measure on the space of Brownian motion paths. Namely, give $0\leq t_1 \leq t_2 \leq \cdot \cdot \cdot \leq t_k$ and Borel sets $E_j \subset \mathbb{R}^{n}$, and starting at $x=0$ and $t=0$, we have
+$$Pr[\omega(t_1) \in E_1, \omega(t_2) \in E_2, \cdot \cdot \cdot,\omega(t_k) \in E_k] = \int_{E_1} \cdot \cdot \cdot \int_{E_k} p(t_k - t_{k-1}, x_k - x_{k-1}) \cdot \cdot \cdot p(t_1, x_1) dx_k \cdot \cdot \cdot dx_1$$
+One last thing before we construct the measure is we have to define the space of Brownian motion paths. We characterize the path by its location at positive rational time t, and the space of all paths is
+$$\mathcal{P} = \prod_{t \in \mathbb{Q}^{+}} \dot{\mathbb{R}}^{n}$$
+where $\dot{\mathbb{R}}^{n} = \mathbb{R}^{n} \cup \{ \infty \}$ is the one-point compactification of $\mathbb{R}^{n}$. Hausdorff's theorem (Wright 1994) ensures that $\mathcal{P}$ is compact. Now, we are ready to construct our measure, motivated by the following theorem:
+==== Theorem 1 ====
+If $X$ is a compact metric space and $\alpha$ is a positive linear functional on $C(X)$, then there exists a unique finite, positive Borel measure $\mu$ such that
+$$\alpha(f) = \int f d\mu$$
+for all $f \in C(X)$
+**proof:** see Taylor (2006) Theorem 13.5\\
+To use this theorem, we construct a positive linear functional $E: C(\mathcal{P}) \rightarrow \mathbb{R}$. We first define $E$ on the subspace $C^{\#}$ with only continuous functions that depend on only finitely many of the factors in $\mathcal{P}$, with the form
+$$\phi (\omega) = F(\omega(t_1), ... , \omega(t_k)) \text{ , } t_1 < ... <t_k$$
+where $F$ is continuous on $\mathcal{P}$ and $t_j \in \mathbb{Q}^{+}$. Then we let
+$$E(\phi) = \int \cdot \cdot \cdot \int p(t_1, x_1) p(t_2 - t_1, x_2 - x_1) \cdot \cdot \cdot p(t_k - t_{k-1}m x_k - x_{k-1}) F(x_1, ..., x_k) dx_k \cdot \cdot \cdot dx_1$$
+To check this functional is well-defined, we resort to the following proposition:
+==== Proposition 1 ====
+For $t \neq s$, $\int p(t, x-y)p(s,y)dy = p(t+s, x)$\\
+**proof:** see Sternberg (2014) slides 10 and 11\\
+Proposition 1 showed that if $F$ does not depend on a given set of $x_i$, we obtain the same functional $E(\phi)$. In addition, we have $E(1) = 1$, so by the Stone-Weierstrass Theorem, $C^{\#}$ is dense in $C(\mathcal{P})$ (see Taylor 2006 Theorem A.23). Furthermore, this functional can be extended to $C(\mathcal{P})$. Therefore, by Theorem 1, we have a measure on the space $\mathcal{P}$:
+==== Theorem 2 (Wiener Measure) ====
+There exists a unique Borel measure (called the Wiener Measure) $\mu$ on $\mathcal{P}$ such that
+$$ E(\phi) = \int_{\mathcal{P}} \phi(\omega) d \mu(\omega)$$
+for each $\phi(\omega)$ with a continuous $F$ on $\mathcal{P}$
+==== References ====
+Sternberg, Shlomo Z. 2014. “Wiener Measure.” Harvard Math 201a, November 11.\\
+\\
+Taylor, Michael E. 2006. Measure Theory and Integration. Graduate Studies in Mathematics, v. 76. Providence, R.I: American Mathematical Society.\\
+\\
+Wright, David G. 1994. “Tychonoff’s Theorem.” Proceedings of the American Mathematical Society 120 (3): 985–87. https://doi.org/10.1090/S0002-9939-1994-1170549-2.

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