Despite these 6 questions
It was pointed out to me in maths educator se that...
Actually, $P(A|B)=1$ does not imply $P(A)=1$ because $0 < P(A=B) < 1$.
In fact I believe '$P(A|B)=1$' is equivalent to '$A \subseteq B$ a.s.' i.e. '$A^c \cap B = \emptyset$ a.s.' meaning $P(A^c \cap B)=0$ (where events are almost surely equal or subset/superset if the corresponding indicator random variables are almost surely, resp, $=$ or $\le$/$\ge$). Right?
What's going on? NNT is wrong? Or NNT is right because e is referring to future information and replacing $B$ with $\Omega$ is actually the basic probability version of coarser partitions in filtrations in advanced probability i.e. past info instead of finer partitions in filtrations i.e. future info?