I don't understand completely the definition of $\sigma$-algebra in the probability triple $(\Omega,\mathcal F, \Bbb P)$ . By the definition, $\mathcal F$ is the set of all events(outcomes) of the sample space $\Omega$. So basically the power set of $\Omega$. Now my concern is: How is the empty set an outcome? For example our sample space is of the experiment of tossing a coin twice. Then $\Omega={HH,HT,TH,TT}$ . How is the empty set an "outcome" ? Also, why is $\mathcal F$ defined as the power set of $\Omega$? Why can't it basically be equal to $\Omega$? Since there is no outcome ${{HH,TT}}$ (since this would be an outcome of the experiment of tossing two coins twice?
Clarifying the meaning of $\mathcal F$ in the probability triple $(\Omega, \mathcal F, \Bbb P)$
definitionprobabilityprobability theory
Best Answer
“Outcome” usually refers to an element of $\Omega$, whereas an event is an element of $\mathcal F$. The outcome is a full description of what happened. An event is something that can be decided based on this full description.
For instance the event that there is at least one head and one tail would be represented by $\{HT,TH\}$. So an event is said to occur if the outcome that occurs is an element of the event. The empty set is a boring event that is characterized by the fact that it never occurs. Similarly $\Omega$ always occurs.
Also note that while $\mathcal F\subset P(\Omega),$ we often want a smaller sigma algebra than the full power set, since the measure we are interested in can’t be defined on the full power set (as in the case of measures on the real line). In the case where $\Omega$ is finite the full power set is the natural choice, though.