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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