Exam
6210-2024-01
Spring 2024 · 6 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Let \(X\) be a set and \(\mu ^*\) be an outer measure on \(X\). A set \(A\subset X\) is called \(\mu ^*\)-measurable if for every \(S\subset X\) we have
\[ \mu ^*(S)=\mu ^*(S\cap A)+\mu ^*(S\setminus A). \]Show that \(\mu ^*\) is countably additive on the \(\mu ^*\) measurable sets.
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Question 2
Prove that there exists \(\phi :\ell ^2(\mathbb {N})\to \mathbb {R}\) that is linear and not continuous.
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Question 3
Show that the set of real numbers that have infinitely many 2s in their base 10 expansion is a Borel set.
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Question 4
Prove that if \(A:\ell ^2(\mathbb {N})\to \ell ^2(\mathbb {N})\) is continuous and linear and \(AB(0,1)\) is dense in \(B(0,1)\) then \(A\) is surjective. Note that this is true for Banach spaces as well (though we are not asking you to prove it).
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Question 5
We say \(f_1,\ldots \) is \(\lambda \)-Cauchy if for every \(\epsilon >0\) there exists \(N\) so that for \(i,j>N\) we have
\[ \lambda (\{x: |f_i(x)-f_j(x)|>\epsilon \})<\epsilon . \]Show that if \(f_1,\ldots \in L^1(\lambda )\) is \(\lambda \)-Cauchy then there exists \(g\in L^1(\lambda )\) so that \(f_1,f_2,\ldots \) converges in measure to \(g\).
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Question 6
Let \((X,\mathcal {M})\) be a measurable space and \(\mu \) and \(\nu \) be two finite measures on \((X,\mathcal {M})\). Show that the following are equivalent
- \(\nu \ll \mu \)
- Whenever \(A_1,\ldots \in \mathcal {M}\) satisfy \(\lim _{n\to \infty }\mu (A_n)=0\) we have \(\lim _{n\to \infty }\nu (A_n)=0\).