The things you think of while doing the dishes. At the Maths Faculty in Cambridge, there are cunningly no dishwashers. Allegedly there are showers, a proven idea-generating tool. But I think they are there for other reasons.
Intuitively, it feels as if “after the fact” the probability of any event should either be 0 or 1. Either it happened or it didn’t. Well, except if your last name is Clinton.
Framing this in terms of rigorous axiomatic probability theory: consider the filtered probability space . Consider real-valued random variables on this space and use the notion of conditional expectations. Define conditional probability in terms of conditional expectation as follows: for and a sub--algebra we have .
Now take – we are thinking of as being the current time, and we are looking at the event which occurred at or before the current time. Since is -measurable, is an -measurable random variable. Thus we can “take out what is known”: .
But indicator functions are always either 0 or 1. Hence the result.
This is a (very, very simple) 01 law. And you don’t really need maths for it.