Axiom Of Choice

Description: A proof that Form 287 (the Hahn-Banach theorem for separable normed linear spaces) implies Form 222 (there is a non-principal measure on $\omega$ ).

Content:

A proof that Form 287 (the Hahn-Banach theorem for separable normed linear spaces) implies Form 222 (there is a non-principal measure on $\omega$ ). This result and its proof are due to M. Morillon.

Let $E$ be the vector space of bounded sequences of reals. Let $M$ be a norm on $E$ such that the normed vector space $(E,M)$ is separable: for example, if $x=(x_{k})_{k\in\Bbb N}\in E$ let $M(x) = {\displaystyle \sum^{}_{k\in {\Bbb N}}{\mid x_{k}\mid \over (k+1){ }^{2}}}$ . Here, the normed vector space $(E,M)$ is separable because the set of sequences in ${\Bbb Q}^{({\Bbb N})}$ which end in an infinite sequence of zeros is dense in E.

For every $x=(x_{k})_{k\in {\Bbb N}}\in E$ , let $p(x)=\overline{\lim}\{x_{k};k\in {\Bbb N}\}$ . Hence, we obtain a sublinear functional $p:E \rightarrow {\Bbb R}$ . Using Form 287, let $g:E \rightarrow{\Bbb R}$ be a linear functional such that $g\le p$ . We have $p(1)=1$ and $p(-1)=-1$ . Hence, $g(1)=1$ . For every finite subset $A\subseteq{\Bbb N}$ we have $p(1_{A})=p(-1_{A})=0$ , hence, $g(1_{A})=0$ . Here $1_A$ denotes the characteristic function of the set $A$ .

For every $A\subseteq{\Bbb N}$ , lf $m(A)=g(1_{A})$ , then $m:{\cal P}(\omega)\rightarrow [0,1]$ is a non-principal measure on $\omega$ .

Howard-Rubin number: 145

Type: Theorem

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