6210-2025-01

Prove the following variant of Egorov’s theorem that applies to sets of infinite measure. Let \((X,\mathcal {M},\mu )\) be a measure space and let \(g\in L^1(\mu )\). Suppose \(f_n:X\to \mathbb {R}\) are measurable with \(|f_n|\le g\) and \(f_n\to f\) pointwise. Then for every \(\epsilon >0\) there is a set \(E\subset X\) of measure \(<\epsilon \) so that \(f_n\to f\) uniformly on \(X\setminus E\).

Let \(\mu \) be a Borel measure on \(\mathbb {R}\) (i.e. a measure defined on the Borel \(\sigma \)-algebra). Suppose it satisfies:

1. \[ \mu ([0,1])=1. \]
2. \(\mu (A+t)=\mu (A)\) for every Borel set \(A\) and every \(t\in \mathbb {R}\), i.e. \(\mu \) is translation invariant.

Show that \(\mu \) equals the Lebesgue measure.

Construct a compact subset in \(\mathbb {R}^2\) with empty interior and with positive Lebesgue measure.

Let \(X\) be a set and \(S\subset \mathcal {P}(X)\) a collection of subsets of \(X\) containing the empty set and the whole space. Let \(\psi :S\to [0,\infty ]\) be a function such that \(\psi (\emptyset )=0\). Define \(\psi ^*:\mathcal {P}(X)\to [0,\infty ]\) as \[ \psi ^*(A)=\inf \left \{\sum _{i=1}^{\infty }\psi (S_i): A\subseteq \bigcup _{i=1}^{\infty }S_i\right \}. \]

Define what it means to be an outer measure and show that \(\psi ^*\) is an outer measure.

Let \(V,W\) be Banach spaces, and let \(T_i:V\to W\) be a sequence of continuous linear operators such that \(\lim _{i\to \infty }T_i v\) exists for every \(v\in V\). Define the linear map \(T:V\to W\) by \[ Tv=\lim _{i\to \infty }T_i v \]

Show that \(T\) is continuous.

Let \(\mathcal {B}\) be the Borel \(\sigma \)-algebra on \(\mathbb {R}\) and \(\mathcal {M}\) be a \(\sigma \)-algebra on \(\mathbb {R}\). Prove that \(f:(\mathbb {R},\mathcal {M})\to (\mathbb {R},\mathcal {B})\) is measurable for all monotone functions \(f\) iff \(\mathcal {M}\) contains the Borel \(\sigma \)-algebra.