(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let D be the collection of all finite subsets ( including the empty set) of [0,1].

Prove that D is a semi-ring. What is [tex]\sigma(D)[/tex] ? Define on D: [tex]\mu (A)[/tex]=#A . Prove that [tex]\mu[/tex] is a premeasure and identify [tex]\mu_{e}[/tex] and

[tex]\Sigma_{mu_{e}}[/tex] . Is ([0,1],[tex]\sigma (D)[/tex], [tex]\mu_{e}[/tex]) complete?

Prove that ([0,1],[tex]\sigma (D)[/tex], [tex]\mu_{e}[/tex]) [tex]\neq[/tex]

([0,1],[tex]\Sigma_{mu_{e}}[/tex],[tex]\mu_{e}[/tex]).

2. Relevant equations

[tex]\mu_{e}[/tex] is the outer measure,

[tex]\Sigma_{mu_{e}}[/tex] is the collection of all [tex]\mu_{e}[/tex] measurable sets.

[tex]\sigma (D)[/tex] is the sigma algebra generated by D

3. The attempt at a solution

showing that D is a semi ring is clear.

but [tex]\sigma (D)[/tex] is a little unclear, since it must be closed under complementation, so if A [tex]\in[/tex] D, then A is a finite set, but A[tex]^{c}[/tex]

may not be a finite set.

showing that [tex]\mu[/tex] is a pre-measure is clear.

any comments for the rest is appreciated.

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# Homework Help: Measure theory

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