Proving D is a Semi-Ring and Identifying \sigma (D) in [0,1]

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Homework Statement



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]).

Homework 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

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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Discussion

Um, the complement of a finite set in [0,1] will definitely not be finite. But why is that a problem? Doesn't it just make the sigma algebra generated by D the collection of all sets with finite complement in [0,1] and their complements?

Edit: Oh, sorry, it has to be closed under countable unions, so I guess it's not that simple.
 
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Why stick to finiteness? sigma algebras work well with countability. The sigma algebra generated by D certainly contains all countable sets and sets whose complement is countable (i.e. cocountable sets); can it contain anything else?