For example, I had no idea that probabilities of 0 or 1 don't change with information. I don't recall being taught this in elementary or advanced probability. The advanced probability way of proving it would be with filtrations,
$$P(A|\mathscr F_t) = 1 \iff P(A|\mathscr F_s) =1 \iff P(A)= 1 \iff P(A|\mathscr G)= 1,$$
but there's a way without measure theory to prove a weaker result in elementary probability:
$$P(A|B) = 1 \iff P(A)=1$$
This needs no measure theory!
If I were unsure about the last statement or had no idea about the weaker result and were to ask something like 'How do I/Can I teach the concept that probabilities of 0 or 1 do not change when you add or remove conditions without measure theory?', what tag is or could be (is: it exists, so what is it? / could be: it doesn't exist, so what could be an appropriate tag?) appropriate?