# Is an event a subspace of the sample space?

In a lecture today, a professor of mine described an event as being "in" the sample space. When writing on the board, for a sample space $S$ and event $E$, it was denoted: $$E \in S$$ This confused me, as I have always thought that events were subsets of the sample space, in which I case I would write: $$E \subset S$$

When I asked after class, I was told that events are not subsets of the sample space. If they are not subsets of the sample space, then how are they defined?

For example, let $S$ be the 6 possible outcomes of rolling a 6-sided die. If we were interested in event $E$, where the number of pips is even, would $E$ not be a subset of all possible outcomes?

• Unless your professor is using unusual definitions of "event" and "sample space," s/he is plainly wrong. (Even Wikipedia is unambiguously clear about that.) Perhaps you could quote the definitions they are using? – whuber Aug 23 '18 at 13:14
• We are using standard definitions of events and sample spaces. A sample space is defined as all possible outcomes of an experiment, and an event is defined as some set of outcomes in the sample space. – James Otto Aug 25 '18 at 1:07
• And that answers your question definitively. – whuber Aug 25 '18 at 13:16

1. An event $E$ is a subset of $S$, however it is an element of sigma-field or sigma-algebra generated by $S$. Perhaps he wrote something like $E \in \sigma(S)$. This is because the sigma-field is a set of sets.
2. I think I recall that certain textbooks differentiate between events and "simple events." In the case of your dice example, $1 \in \{1,2,3,4,5,6\}$, but $\{1\} \subset \{1,2,3,4,5,6\}$. In the first case, a simple event is an element of the space, and in the second, it's a set. I don't know, though. I find this confusing myself.
• I should have mentioned in my original post, I asked my professor after class and they were not referring to the sigma-algebra generated by $S$. I have never heard of "simple events", that is very interesting. However, the events we were discussing had more than one element, so I do not believe they could be classified as "simple events", and would need to be subsets of $S$. – James Otto Aug 25 '18 at 1:00