Elementary Properties of Event Space

Theorem
Let $$\mathcal E$$ be an experiment with a probability space $$\left({\Omega, \Sigma, \Pr}\right)$$.

The event space $$\Sigma$$ of $$\mathcal E$$ has the following properties:


 * 1) $$\varnothing \in \Sigma$$;
 * 2) $$\Omega \in \Sigma$$;
 * 3) $$A, B \in \Sigma \implies A \cap B \in \Sigma$$;
 * 4) $$A, B \in \Sigma \implies A \setminus B \in \Sigma$$;
 * 5) $$A, B \in \Sigma \implies A \ast B \in \Sigma$$;
 * 6) $$A_1, A_2, \ldots \in \Sigma \implies \bigcap_{i=1}^\infty A_i \in \Sigma$$, that is, the intersection of any countable collection of elements of $$\Sigma$$ is also in $$\Sigma$$.

In the above:
 * $$A \setminus B$$ denotes set difference;
 * $$A \ast B$$ denotes symmetric difference.

Proof
By definition, a probability space $$\left({\Omega, \Sigma, \Pr}\right)$$ is a measure space.

So, again by definition, an event space $$\Sigma$$ is a sigma-algebra on $$\Omega$$. Thus the requirements above.

As $$\Sigma$$ is a sigma-algebra, it is also by definition an algebra of sets.

It follows from Properties of Algebras of Sets and Equivalence of Definitions of Algebra of Sets, that:
 * $$\varnothing \in \Sigma$$;
 * $$\Omega \in \Sigma$$;
 * $$A, B \in \Sigma \implies A \cap B \in \Sigma$$;
 * $$A, B \in \Sigma \implies A \setminus B \in \Sigma$$;
 * $$A, B \in \Sigma \implies A \ast B \in \Sigma$$.


 * Finally, note that as every sigma-algebra is also a delta-algebra:
 * $$A_1, A_2, \ldots \in \Sigma \implies \bigcap_{i=1}^\infty A_i \in \Sigma$$

by definition of delta-algebra.