[PATCH] Updated
Sean Farley
sean.michael.farley at gmail.com
Mon Feb 24 21:55:22 CST 2014
Francisco Figueirido <francisco at giantclam.net> writes:
> # HG changeset patch
> # User francisco <francisco at giantclam.net>
> # Date 1393296985 18000
> # Node ID 317d7cb86c8f51156ed5cf966c4fc0a7e28b323f
> # Parent 129b7f5b99664b38e251dd2d64a822f8e9c2f8db
> Updated
This patch applies to my mercurial repo a.e. since the diff is finite
and, therefore, measure zero. Unfortunately, though, I don't think the
mercurial devs are as happy as I am to discuss measure theory ;-)
> diff -r 129b7f5b9966 -r 317d7cb86c8f rw2.tex
> --- a/rw2.tex Mon Feb 24 00:22:13 2014 -0500
> +++ b/rw2.tex Mon Feb 24 21:56:25 2014 -0500
> @@ -9089,13 +9089,53 @@
> \]
> for any non-negative $t$ (the case where $t=0$ is trivially true).
>
> -Notice that if $x\in A$ then $p_A(x)\le 1$.
> -
> -Now consider a point $x\not\in A$ and let $\YY\subset\XX$ be the one-dimensional subspace generated by $x$. Take any $0<\alpha<p_A(x)$ and define the linear functional
> -\[
> -\phi_A( t\,x ) = t\,\alpha \le \abs{t}\,\alpha < p_A( t\, x ).
> -\]
> -The Hahn-Banach Theorem then implies that $\phi_A$ can be extended to the whole of $\XX$ maintaining the bound.
> +Notice that:
> +\begin{itemize}
> +\item If $x\in A$ then $p_A(x)\le 1$.
> +\item If $0\in A$ for any $x\in A$ and any $s>1$ we have that $x/s = x/s + (1-1/s)\,0 \in A$.
> +\item If $0\in A$ and $x\not\in A$ then $x/s\in A$ implies that $s>1$ for if $x/s\in A$ for some $s<1$ we would then have, by the above and choosing $t=1/s>1$, that $x=(x/s)/t\in A$ which is a contradiction. This implies that $p_A(x)\ge 1$ for all $x\not\in A$.
> +\item If $\XX$ is a topological vector space, $0\in A$ and $x\not\in\closure{A}$ then there is some $\epsilon>0$ such that $x/(1+h)\not\in A$ for any $0\le h\le\epsilon$ and therefore $p_A(x)\ge 1+\epsilon>1$. Conversely, if $p_A(x)=1$ then there is some sequence $s_n$ converging from above to $1$ such that $x/s_n\in A$ and therefore its limit, $x$, is in $\closure{A}$. We thus conclude that $\closure{A}=\{ x : p_A(x) \le 1 \}$.
> +\end{itemize}
> +Assume $0\in A$ and consider a point $x\not\in A$; let $\YY\subset\XX$ be the one-dimensional subspace generated by $x$. Since $p_A(x)\ge 1$ we can define the non-zero linear functional
> +\[
> +\phi_A( t\,x ) = t\,p_A(x) .
> +\]
> +If $t<0$ we clearly have $\phi_A(t x) < 0 \le p_A( t x )$; and when $t\ge 0$ we have $\phi(t x) = t p_A(x) = p_A(t x)$, so the Hahn-Banach Theorem allows us to extend $\phi_A$ to a linear functional defined on the whole $\XX$ space, call it $\phi$, that satisfies the bound $\phi(u)\le p_A(u)$ for any $u\in \XX$. For this extension we then have
> +\begin{itemize}
> +\item $x\in \{ y : \phi(y)\ge 1\}$.
> +\item If $x\not\in\closure{A}$ then $x\in \{ y : \phi(y) > 1\}$.
> +\item $A \subset \{ y : \phi(y) \le 1 \}$
> +\end{itemize}
> +so that the (convex) space defined by the condition $\{ y : \phi(y) = 1\}$ separates $A$ from $x$ (although $x$ could be in the intersection).
> +
> +\noindent For any linear functional like $\phi$ one defines the kernel of $\phi$ as
> +\[
> +\Kernel(\phi) = \bigl\{ z : \phi(z) = 0 \bigr\}.
> +\]
> +Notice that the kernel of $\phi$ has codimension 1: take any point $x$ not in the kernel and consider any other point $y\in\XX$. Then we have $\phi(x)\ne 0$ and therefore
> +\[
> +u = y - \dfrac{\phi(y)}{\phi(x)} x \in \XX\quad\text{and}\quad \phi(u) = 0
> +\]
> +so that $y = u + (\phi(y)/\phi(x)) x$ which proves that any point in $\XX$ can be written as the sum of a point $u$ in the kernel of $\phi$ plus a point in the one-dimensional linear space generated by $x$.
> +Moreover, clearly for any two points $z_1$ and $z_2$ such that $\phi(z_1)=\phi(z_2)=1$ the difference $z_1-z_2\in\Kernel(\phi)$ and therefore if we choose some arbitrary point $z_1$ with $\phi(z_1)=1$ we can write
> +\[
> +\bigl\{ z : \phi(z) = 1 \bigr\} = \bigl\{ w + z_1, \, w\in\Kernel(\phi) \bigr\} = T_{z_1}\left( \Kernel(\phi) \right)
> +\]
> +where $T_u$ is the translation operator $T_u(y) = y + u$, which is essentially the definition of a hyperplane.
> +
> +Suppose now that $\XX$ is a normed space (see Appendix~\ref{ap:normed-space}) with norm $x\rightarrow\norm{x}$ and $x\rightarrow p_A(x)$ is the Minkowski functional of a convex set $A$ that contains $0$. Since
> +\[
> +p_A\left( \dfrac{x}{\norm{x}} \right) = \dfrac{1}{\norm{x}}\, p_A(x) \Rightarrow p_A(x) = \norm{x}\, p_A\left( \dfrac{x}{\norm{x}} \right)
> +\]
> +we can conclude that the extension of a bounded functional will be continuous if
> +\[
> +\sup_{u : \norm{u}=1} p_A(u) < \infty.
> +\]
> +Indeed, in this case we have, for any $x\in\XX$,
> +\begin{align*}
> +\phi(x) \ge 0 & \Rightarrow \abs{\phi(x)} = \phi(x) \le p_A(x) \le \left( \sup_{u : \norm{u}=1} p_A(u) \right)\,\norm{x} \\
> +\phi(x) < 0 & \Rightarrow \abs{\phi(x)} = -\phi(x) = \phi(-x) \le p_A(-x) \le \left( \sup_{u : \norm{u}=1} p_A(u) \right)\,\norm{x}
> +\end{align*}
>
> \chapter{Radon-Nikodym Theorem\\Conditional Expectation}
> \label{ap:radon-nikodym}
> diff -r 129b7f5b9966 -r 317d7cb86c8f rw2mac.tex
> --- a/rw2mac.tex Mon Feb 24 00:22:13 2014 -0500
> +++ b/rw2mac.tex Mon Feb 24 21:56:25 2014 -0500
> @@ -213,6 +213,8 @@
> %% Bold symbol macro for standard LaTeX users
> \providecommand{\boldsymbol}[1]{\mbox{\boldmath $#1$}}
>
> +\def\Kernel(#1){\textsf{Kern}\left({#1}\right)}
> +
> \bibliographystyle{alpha}
>
> \makeatother
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