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## Homework Statement

Prove that the σ-algebra generated by the collection of all intervals in R

^{n}coincides with the σ-algebra generated by the collection of all open subsets of R

^{n}.

## 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 R

^{n}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 R

^{n}contains all subsets of R

^{n}including ∅.

2. The σ-algebra generated by the collection of all intervals in R

^{n}contains all intervals of R

^{n}including ∅. Since an interval is a subset, every element of the σ-algebra generated by the collection of all intervals in R

^{n}would coincide with the σ-algebra generated by the collection of all open subsets in R

^{n}.

How far off am I? Is there a better way to make this argument?