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ChemEng1
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Homework Statement
Prove that the σ-algebra generated by the collection of all intervals in Rn coincides with the σ-algebra generated by the collection of all open subsets of Rn.
Homework Equations
A σ-algebra is a nonempty collection Σ of subsets of X (including X itself) that is closed under the complement and countable unions of its members.
The Attempt at a Solution
1. The σ-algebra generated by the collection of all open subsets of Rn would also contain all closed subsets by complementation. It would also contain their unions to generate all partly open subsets. Therefore the σ-algebra generated by the collection of all open subsets of Rn contains all subsets of Rn including ∅.
2. The σ-algebra generated by the collection of all intervals in Rn contains all intervals of Rn including ∅. Since an interval is a subset, every element of the σ-algebra generated by the collection of all intervals in Rn would coincide with the σ-algebra generated by the collection of all open subsets in Rn.
How far off am I? Is there a better way to make this argument?