Exercise

Functional Analysis
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Papapetros Vaggelis
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Exercise

#1

Post by Papapetros Vaggelis »

Let \(\displaystyle{\left(X,||\cdot||\right)}\) be an infinite dimensional normed space over \(\displaystyle{\mathbb{R}}\) .

Prove that there does not exist a \(\displaystyle{\rm{Borel}}\) measure \(\displaystyle{\mu}\) on \(\displaystyle{X}\)

such that

1. \(\,\,\,\,\displaystyle{\mu(x+A)=\mu(A)\,,\forall\,A\in\mathbb{B}(X)\,\,,\forall\,x\in X}\)

2. \(\,\,\,\,\displaystyle{\mu(A)>0}\) for every non empty and open subset of \(\displaystyle{X}\)

3. there exists a non empty open subset \(\displaystyle{A_{0}}\) of \(\displaystyle{X}\) with \(\displaystyle{\mu(A_{0})<\infty}\) .
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