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:
 * $$\varnothing \in \Sigma$$;
 * $$\Omega \in \Sigma$$;
 * $$A, B \in \Sigma \implies A \cap B \in \Sigma$$;
 * $$A, B \in \Sigma \implies A \setminus B \in \Sigma$$;
 * $$A, B \in \Sigma \implies A \ast B \in \Sigma$$.

Now, an event space $$\Sigma$$ has the following properties by definition:


 * $$A \in \Sigma \implies \mathcal{C}_\Omega \left({A}\right) \in \Sigma$$, that is, the relative complement of any element of $$\Sigma$$ is also in $$\Sigma$$;


 * $$A_1, A_2, \ldots \in \Sigma \implies \bigcup_{i=1}^\infty A_i \in \Sigma$$, that is, the union of any countable collection of elements of $$\Sigma$$ is also in $$\Sigma$$.

Hence we have that $$\Omega \setminus \bigcup_{i=1}^\infty A_i \in \Sigma$$, where $$\setminus$$ denotes set difference.

But by De Morgan's laws, $$\Omega \setminus \bigcup_{i=1}^\infty A_i = \bigcap_{i=1}^\infty A_i$$.

Hence the result $$A_1, A_2, \ldots \in \Sigma \implies \bigcap_{i=1}^\infty A_i \in \Sigma$$.