# Homework 10, Real Analysis 2

## Due May 4

### Problem 1

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

### Problem 2

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

1. $\overline{\mathcal M}$ is the smallest $\sigma$-algebra containing $\mathcal M$ and all subsets of its elements of measure 0.
2. The function $\bar\mu$ is a complete measure on $\overline{\mathcal M}$.

### Problem 3

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

### Problem 4

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

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