Unformatted text preview: The sets in the list above include R , which is of the second form with a =-∞ . “Finite unions” include unions with no sets at all, putting the empty set into the mix, and unions involving only one set. So A contains R , ∅ , all intervals of the form ( a,b ], all intervals of the form ( a, ∞ ), all unions of two sets of the type just mentioned, all unions of three such sets, etc. Prove A is an algebra. Hint: The only difficult part (i.e., non-trivial part) is to see that A ∈ A implies A c ∈ A . For this purpose you should prove first: If A,B ∈ A , then A ∩ B ∈ A . Once you have this, you can proceed by induction. Let us call the sets listed in I,II basic sets. Go by induction on n , proving: If A is a union of n basic sets, then A c ∈ A . But if you find a better way, by all means use it. 4...
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- Spring '11
- Sets, Empty set, measure, Basic concepts in set theory, Lebesgue measure