Set Difference 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:

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

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

$\ds A, B$

$\in$

$\ds \Sigma$

$\ds \leadsto \ \ $

$\ds A, \Omega \setminus B$

$\in$

$\ds \Sigma$

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

$\ds \leadsto \ \ $

$\ds A \cap \paren {\Omega \setminus B}$

$\in$

$\ds \Sigma$

$\ds \leadsto \ \ $

$\ds A \setminus B$

$\in$

$\ds \Sigma$

$\blacksquare$

1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events: Exercise $2$
1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.4$: Probability spaces: $(12)$