Exercise
Posted: Wed Nov 11, 2015 4:02 pm
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 \(\displaystyle{\mu(A_{0})<\infty}\) .
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 \(\displaystyle{\mu(A_{0})<\infty}\) .