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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