Question on the $\sigma$-Algebra of a Given Probability Space

Question

I am trying to understand what a sigma algebra is and I found this really helpful pdf online. On the last couple of lines on page 1 though, the author writes: "Any event E is a subset of Ω. Assume that the set of all events are represented by a particular family of sets over Ω denoted A, i.e. A ⊆ P(Ω)". Here's my reasoning: We know that E is a set that contains outcomes of our experiment with some specific property. So, it is true that: $$ \forall x \in Ε \Rightarrow x \in \Omega \ or \ Ε \subseteq \Omega $$ We should expect then the family of all events in Ω to be the set: $$ A= \{Ζ|Ζ \subseteq \Omega \} $$ as the property $ Z \subseteq Ω $ defines an event. The second side of the equation above is what is called the power set of Ω: $$ P(Ω)=\{Ζ|Ζ \subseteq Ω \} $$ Therefore: $ A=P(Ω) $. Ι don't get why the author writes $ A \subseteq P(Ω) $ which leaves the possibility for A to be a proper subset of $ P(Ω) $. Shouldn't it be strictly $A=P(\Omega)$?

Answer 1

The author writes this because in general not every set in the $\sigma$-algebra is measurable, that is, can be assigned a probability. Therefore, not all sets in the $\sigma$-algebra can be "lawfully" termed "events".