I'm pretty sure this has a trivial answer but it's always faster to ask the community :-)
I understand that, relative to a random oracle, P=BPP. But this is sometimes phrased via the shorthand "Almost-P=BPP", which I find rather confusing.
Complexity zoo defines Almost-P as "The class of problems that are in $P^A$ with probability 1, where A is an oracle chosen uniformly at random."
Logically, L belonging to Almost-P would then mean that: With probability 1 over a random instantiation of an oracle tape A, there is a polynomial time Turing machine T(A) that decides L.
Now, the poor BPP machine trying to decide L would have no idea which T(A) to simulate. The choice of T(A) may wildly vary depending on A.
Should Almost-P (similarly, Almost-anything) be rather defined as the set of languages L for which there is a polynomial time Turing machine T with an oracle tape such that, with probability 1 over the instantiation of the oracle tape, the language decided by T ends up being L?
Under neither of the above definitions does the statement BPP=Almost-P seems to be logically equivalent to "relative to a random oracle A, $P^A=BPP^A$".
My question is now, what is the precise definition of the class Almost-P (or, for that matter, Almost-anything)? And/or, what am I missing above?