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:


 * $(1) \quad \varnothing \in \Sigma$
 * $(2) \quad \Omega \in \Sigma$
 * $(3) \quad A, B \in \Sigma \implies A \cap B \in \Sigma$
 * $(4) \quad A, B \in \Sigma \implies A \setminus B \in \Sigma$
 * $(5) \quad A, B \in \Sigma \implies A \ast B \in \Sigma$
 * $(6) \quad A_1, A_2, \ldots \in \Sigma \implies \displaystyle \bigcap_{i=1}^\infty A_i \in \Sigma$, 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:
 * $\displaystyle A_1, A_2, \ldots \in \Sigma \implies \bigcap_{i=1}^\infty A_i \in \Sigma$

by definition of delta-algebra.