[srean]

Sure you can. The TLDR would be "piecewise constant projection"

I think picking up a standard graduate probability book will clear this up better than any long comment trail. There are no problems defining a coarser sigma algebra using an original one and then defining a function measurable on the new sigma algebra. Note this continues to be an r.v. in the original space as meaurability is preserved. A consistent definition the values of the conditioned r.v. would be the piecewise constant approximation of the original r.v. over the indivisible elements of the coarser sigma algebra.

Let me try another route.

You seem to be accepting of a conditional expectation. Now what is a conditional expectation if not a function. Now all we need is that function be measurable with respect to the new sigma algebra, thats ensured byconstruction. Hope it helped some

[bonoboTP]

> I think picking up a standard graduate probability book will clear this up better than any long comment trail.

Can you recommend one? I just picked up Probability and Measure by Billingsley and it does not mention "conditional random variable" a single time in over 600 pages. It does have a lot of "conditional probability", "conditional distribution", "conditional expectation" etc.

> You seem to be accepting of a conditional expectation.

Conditional expectation is defined in terms of conditional probabilities, and those are in turn explicitly defined as P(A|B)=P(A,B)/P(B), so there's nothing not to accept.

[srean]

Billingsley is pretty darn good. It might have left the connection as a dotted line given that the notion is no different from conditional expectation. The only connection you have to make is conditional expectation is a function and a random variable. You must have seen expectation taken of a conditional expectation. That should should convince you that condititional expectation is indeed a random variable. Since that r.v. was obtained by conditioning its not a stretvh to call it a conditioned r.v.

Any book that explains conditioning over a sigma algebra should suffice. You could try Loeve, Dudely or Neveu but dont remember if its mentioned explicitly.

BTW conditional expectation is really more fundamental than conditional probability. Its the former that yields the latter in measure theoretic probability. If you want to drink from the source that would be Kolmogorov.

Finally if you are reading Billingsley you are adequately qualified to call yourself a mathematician.

[bonoboTP]

It's getting a little tedious. Please show me a concrete citation of a serious textbook (not a tutorial/handout by a grad student or a paper by a random researcher) that puts the three words "conditional random variable" next to each other (consistently, not simply as a one-off potential mistake). Google doesn't show serious sources for it.

While I agree with isolated points of your comment I think it doesn't add up to a useful/coherent concept of conditional random variable.

[srean]

Thats a little too much to ask, perhaps if they were grep'able I could have obliged, unfortunately I dont have a photographic memory.

More concretely its just another name for conditional expectation. I am assuming you are aware that conditional expectation is a random variable obtained via conditioning (equivalently as a piecewise approximation in L_2). If you arent familiar with that view point that would be the place to start. Kolmogorov, Neveu, Dudely, Billingsley will all cover that view point.

[bonoboTP]

> I am assuming you are aware that conditional expectation is a random variable

That's not what we're considering here, but things of the form X|Y=y for a concrete y. Even as E[X|Y=y], that's not a function, y is specified. Do you agree we shouldn't call X|Y=y a conditional random variable?

[srean]

Oh absolutely for a specific y its not function (or a random variable) one usually thinks of Y as a variable and not a constant.