Intersection of Events is Event

Theorem

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.

The event space $\Sigma$ of $\EE$ has the property that:

$A, B \in \Sigma \implies A \cap B \in \Sigma$

That is, the intersection of two events is also an event in the event space.

Proof

\(\ds A, B\)

\(\in\)

\(\ds \Sigma\)

\(\ds \leadsto \ \ \)

\(\ds \Omega \setminus A, \ \Omega \setminus B\)

\(\in\)

\(\ds \Sigma\)

Definition of Event Space: Axiom $(\text {ES} 2)$

\(\ds \leadsto \ \ \)

\(\ds \paren {\Omega \setminus A} \cup \paren {\Omega \setminus A}\)

\(\in\)

\(\ds \Sigma\)

Definition of Event Space: Axiom $(\text {ES} 3)$

\(\ds \leadsto \ \ \)

\(\ds \Omega \setminus \paren {A \cap B}\)

\(\in\)

\(\ds \Sigma\)

De Morgan's Laws: Difference with Intersection

\(\ds \leadsto \ \ \)

\(\ds \Omega \setminus \paren {\Omega \setminus \paren {A \cap B} }\)

\(\in\)

\(\ds \Sigma\)

Definition of Event Space: Axiom $(\text {ES} 2)$

\(\ds \leadsto \ \ \)

\(\ds A \cap B\)

\(\in\)

\(\ds \Sigma\)

Relative Complement of Relative Complement

$\blacksquare$

Also see

• Elementary Properties of Event Space

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events: Exercise $1$