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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?

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“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.

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  • $\begingroup$ Sorry, could you clarify what do you mean by ''So an event is said to occur if the outcome that occurs is an element of the event"? $\endgroup$ – user634512 Jan 11 '20 at 19:46
  • $\begingroup$ Say the event of interest is $\{HT,TH\}$. Then for example, if the outcome is HT then this event does occur, whereas if the outcome is HH, then it does not. $\endgroup$ – spaceisdarkgreen Jan 11 '20 at 19:49
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The elements of $\Omega$ are the so-called elementary events. It's useful and interesting to construct more complex events from them. For example, the event $E_1 =$ 'When tossing the coin twice, we get both times the same result' is in symbols $$ E_1 = \{ HH, TT \}. $$ Note that this is an element of the power set of $\Omega$. Similarly, the event $E_2 =$ 'When tossing the coin twice, we get different results' is in symbols $$ E_2 = \{ HT, TH \} = \Omega \setminus E_2. $$ So you see that sets of elementary events can be used to formally express interesting more complex events.

Also, the empty set is the event that nothing happens. Appropriately, this set has probability zero in any probability space. $$ \mathbb P (\emptyset) = 0. $$ Nevertheless, the empty event is useful, for example $E_1 \cap E_2$ is the event that, when tossing the coin twice, we get at the same time equal and different results. Of course, this does not happen, and appropriately $$ E_1 \cap E_2 = \emptyset, $$ as you can verify from the expressions for $E_1$ and $E_2$ above.

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  • $\begingroup$ Okay, I understand. But physically how is it possible to describe $\emptyset$ ? How is it possible that if I flip a coin twice and nothing occurs? $\endgroup$ – user634512 Jan 11 '20 at 19:40
  • $\begingroup$ In my answer, I give an example of how the empty event "naturally" pops up when we manipulate other events. You can compare it to the number zero, whose existence and meaning were not always as clear as they are nowadays. See here: en.wikipedia.org/wiki/0#Classical_antiquity $\endgroup$ – jflipp Jan 12 '20 at 12:01

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