Exercise
-
- Community Team
- Posts: 426
- Joined: Mon Nov 09, 2015 1:52 pm
Exercise
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 \(\displaystyle{\mu(A_{0})<\infty}\) .
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 1 guest