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Let $ X $ be a set.

Let $ \Sigma $ be a subset of the power set of $ X $

Then, $ \Sigma $ is a σ-algebra on the set $ X $ if the following is true:

  1. $ X \in \Sigma $ ($ X $ is an element of $ \Sigma $.)
  2. $ A \in \Sigma \Rightarrow A^C \in \Sigma $ (For any set, if a set is an element of $ \Sigma $, then its complement is in $ \Sigma $ also.)
  3. $ \{A_i\}_{i \in I} \in \Sigma \Rightarrow \bigcup_{i \in I}A_i \in \Sigma $ (If there is a countable collection of sets that are elements of $ \Sigma $, then the union of those elements are also in $ \Sigma $).

If $ \Sigma $ is a σ-algebra on the set $ X $, then $ (X,\Sigma) $ is a measure space.

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