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--- math105-s22:s:mchlxo:start [2022/05/06 08:24]
mchlxo [Proposition 1]
+++ math105-s22:s:mchlxo:start [2026/02/21 14:41] (current)
@@ Line 75: / Line 75: @@
 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 Theorem A.23). Furthermore, this functional can be extended to $C(\mathcal{P})$. Therefore, by Theorem 1, we have a measure on the space $\mathcal{P}$:
+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.
 ===== Resources =====
 A {{ :math105-s22:s:mchlxo:exception_points.pdf |paper}} that constructs a set which includes points at which the density of the set can take on any values in $[0,1]$

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