Are conditional events sets?

Question

In our probability class, we recently learned that the probability measure, $P$, is a set function that takes in a subset of some sample space, $\Omega$, and returns a numerical value that satisfies the probability axioms. We are now learning about conditional probabilities, namely, $P(A|B)$, for two events $A, B \subset \Omega$. To my understanding, $A|B$ only makes sense in a probabilistic context as "the event that $A$ occurs given that $B$ occurs". However, since the probability measure is a set function, does this mean that $A|B$ is also a set somehow? I am unable to see how this makes sense. Thanks.

EDIT: Thanks for some of the answers so far. If $A|B$ is not a set, then what exactly is it's type? If it isn't a set, why can we take the informality of passing it to a set function?

Answer 1

Good question. The literal answer is "No, $A|B$ is not a set." But there is good intuition behind your question. $A|B$ is not a set, but $A \cap B$ is. Conditional probability defines a probability function on the subsets of the subset (event) $B$ of the universe $\Omega$.

Answer 2

Yes, we can see $A|B$ as a set. You can see it as a subset of the set $B$, it is precisely those elements in the set $B$ which are also in the set $A$. Since $B$ has already happened, we know we are "in" the set $B$. Then, we are looking for the elements in which $A$ also occurs, i.e. the elements which are in $B$ and are also in $A$. However, it is worth noting that $A|B$ is somewhat meaningless if it's not inside a probability measure $\mathbb{P}$. We would just write $A\cap B$.

To answer your edit, it's technically just notation. When we pass $A|B$ into a probability measure we are not simply taking the measure of $A\cap B$, we are asking something more complex, because we are requiring $B$ to have occurred, which is meaningful given the context. Writing $\mathbb{P}(A|B)$ is just notation for "the probability that $A$ occurred given that $B$ has already occurred." Nothing more.