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Homework 10, Real Analysis 2

Due May 4

Problem 1

Let X be a set and $latex \mathcal M$ a nonempty collection of subsets of X closed under complements and countable unions of disjoint sets. Then $latex \mathcal M$ is a $latex \sigma$-algebra.

Problem 2

Let $latex (X,\mathcal M, \mu)$ be a measure space. Its completion is defined as the collection $latex \overline{\mathcal M}$ of sets of the form $latex E\cup N$, where $latex E\in\mathcal M$ and $latex N\subset F$ for some $latex F\in\mathcal M$ with $latex \mu(F)=0,$ and $latex \bar\mu(E\cup N) = \mu(E).$

  1. $latex \overline{\mathcal M}$ is the smallest $latex \sigma$-algebra containing $latex \mathcal M$ and all subsets of its elements of measure 0.

  2. The function $latex \bar\mu$ is a complete measure on $latex \overline{\mathcal M}$.

Problem 3

Consider the Lebesgue exterior measure. Then a set is Caratheodory measurable if and only if is Lebesgue measurable.

Problem 4

Let $latex (X,\mathcal M,\mu)$ be a measure space, $latex A,B,C$ subsets of X such that $latex A\subset B\subset C$, $latex A,C\in\mathcal M$ and $latex \mu(A) = \mu(C)$. Then $latex B\in\mathcal M$.