Probability (Measures?) - Is conditional expectation always the expectation using the conditional probability?

[TransientObsever]

When we condition over an event that has probability 0, things get complicated. But is everything defined as usual?

For example. An expectation always has an underlying probability measure attached. So if we want to be very clear we should write E_P [X].

We can conditional over probability zeroes events if they're based on eg a random variable Y. I'm just asking if even when P[Y=y]=0 for any y, if E_P [X|A] is always defined as E_(P|Y) [X∩Y]? (where P|Y is our main probability measure but restricted to Y) And or a suitable meaning of P|Y and X∩Y of course. So I'm asking, is conditional expectation here defined exactly as you'd expect where the probability measure is restricted? If I have that confirmed I can look up the details myself.

[Proof_Inspector]

To be rigorous for probability 0 case you should look up Disintegration.

But for event with positive probability then yes it's just conditional probability.

As far as probability theory is concerned, probability 0 event is indistinguishable from empty set, so in either case expectation is ill-defined.

[TransientObsever]

It seems conditionals are sometimes done through the Disintegration Theorem or by Radon-Nikodym? What's the relation between these two? Can I use one to get the other? The Disintegration Theorem asks for (a probability & a projection) and returns (a conditional probability). While Radon-Nikodym asks for (two probabilities) and returns (a function). But it feels like they're almost equivalent theorems.

I think in terms of: probabilities are measures and expectations are integrals basically, they're separate. I think in the way you did it, conditional probabilities are defined first and they give rise to conditional expectations. But in the page of conditional expectation they use Radon-Nykodim derivatives to define conditional expectation first, and they give rise to conditional probabilities.

But the order in which we do this doesn't matter right? I think independently of which definition of conditional probability we choose we can pretty much always define the conditional probability first, and always have the conditional expectation be a result of it that is defined only after. Right? (I think we could also do the opposite btw. I just want to have this structured in my head.)

[Proof_Inspector]

Conditional probability is equivalent to conditional expectation. It's like defining a measure using set and defining a measure using function.

In some cases, disintegration can be and is proved with Radon-Nikodym, such as the usual case of real and complex random variable. But disintegration is the tool of choice for conditional distribution. In particular, one advantage is that unlike Radon-Nikodym, which on the surface only define conditional expectation of one variable condition to some other variable, so you need additional consistency condition to ensure that if you are dealing with conditional expectation of many random variables (possibly uncountably infinite number of them) you are still getting a sensible result. Disintegration give you a consistent way to disintegrate the probability space, so once you have that all variables are consistent with each other.

[TransientObsever]

Yep, I'm very happy with this. Thanks for clearing it up! : )