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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