# About event definition (probability)

On Wikipedia, an event is defined as a set of outcomes of an experiment (or in other words, a subset of the sample space of the experiment) right?

In my probability class, we said at some point that:

If $a$ is a $σ$-algebra of a sample space $S$, then any $A \in a$ is an event.

My question is:

This is not the definition of an event right? Because, for example, if $S$ is the sample space of throwing a die, and we have the set $\{ \emptyset, S\}$ which is a $σ$-algebra of $S$, then the set $A=\{1,2\}$, which is not an element of the given $σ$-algebra, is still called an event, right?

Any help to alleviate this confusion would be appreciated.

• It should really be more like "If $a$ is the $\sigma$-algebra of the sample space that we're talking about then we say that any $A \in a$ is an event". That said, we do indeed have situations where there are subsets of our sample space that we don't call events. If for some reason we were talking about the die rolling experiment under the $\sigma$-algebra $\{ \emptyset,\{ 1,2,3,4,5,6 \} \}$ then $\{ 1,2 \}$ wouldn't be an event. – Ian Sep 4 '17 at 16:40
• @Ian So basically, if I was talking about an experiment, i.e. the rolling die, and I haven't defined a specific σ-algebra, then the σ-algebra would be considered as the power set of the sample space and then $A$ would be an event? – Michalis P. Sep 4 '17 at 16:48
• In the probability space that you mention which is equipped with $\sigma$-algebra $\{\varnothing,S\}$ the set $A=\{1,2\}$ is not an event. Of course it can happen that you throw $1$ or $2$ with your die. That means that the chosen probability space is innappropriate to model reality. – drhab Sep 4 '17 at 16:49
• @drhab I'd say that last sentence is overstated; it can happen that you land inside a Vitali set when drawing a number uniformly from $[0,1]$ but no one objects to that. – Ian Sep 4 '17 at 16:52
• @ZeroPancakes In measure-theoretic probability you should always define a specific $\sigma$-algebra. But for countable state spaces it is usually the power set unless otherwise specified, because in this case, specifying a probability measure is the same as specifying $P(\{ x \})$ for each $x$ in the sample space separately. And usually you want to do this anyway. – Ian Sep 4 '17 at 16:52

## 1 Answer

If $a$ is a $σ$-algebra of a sample space $S$, then any $A \in a$ is an event.

No, indeed, that is not defining.   Events are defined as subsets of the sample space.   A $\sigma$-algebra is defined as a set of events, including the empty event, that is closed under countable unions, countable intersections, and relative complementation.

As such, all elements of a $\sigma$-algebra are events, but not all events need be elements of a $\sigma$-algebra.

• In common parlance among probabilists, once we have fixed an underlying $\sigma$-algebra for our sample space, we do not refer to subsets of the sample space outside of this $\sigma$-algebra as being events. Thus in the "canonical" space $([0,1],\mathcal{B}([0,1]),m)$ where $m$ is the 1D Lebesgue measure, a Vitali set is not an event. – Ian Sep 4 '17 at 17:03
• (Cont.) The disconnect between elementary probability and measure-theoretic probability is the reason for this confusion. In elementary probability, we teach a definition that is consistent with common usage only for countable sample spaces, and then don't update the definition when we switch over to uncountable sample spaces (doing so only after switching over to measure-theoretic probability). – Ian Sep 4 '17 at 17:06
• This not in accordance with wikipedia: "So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events" – drhab Sep 4 '17 at 17:16