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.

$\blacksquare$

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.2$
• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.2$: Exercise $1, \ 2, \ 3, \ 4$

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.4 \ (12), \ (13)$