Definition of Condition Expectation( of random variable) can be seen here: http://en.wikipedia.org/wiki/Conditional_expectation

My silly question is: Intuitively, conditional expectation seems to be a restriction of a measurable function on the sub-$\sigma$-algebra. Why don't people call it a restriction instead of such a confusing name. After all, one expects 'expectation' to be a number instead of a function.

Thanks

Modified: Sorry for the vagueness. Here I'm talking about conditional expectation w.r.t a sub-$\sigma$-algebra. And what I mean by 'restriction' is the following:

A random variable is a measurable function defined on the whole sample space. Once defined, it becomes definite.

However if we don't know how this function is defined, there is another way to collect information of it by observing how it works on the events. More precisely, we observe the integral of this function on each set in the corresponding $\sigma$ algebra. After we complete the observation on all sets, we have a good idea on this function(random variable).---But I don't know if we know everything about it?

With the above point of view, the conditional expectation w.r.t a sub-$\sigma$-algebra seems to finish a observation on some but not all sets in the original $\sigma$-algebra. In that sense, it is a restriction.

I want to know if this intuition is correct.

isa number, more precisely, afamilyof numbers. An inclusion of sigma-algebras can be thought of geometrically as a fibration of measurable spaces and a conditional expectation takes a function on the total space, integrates it fiberwise and produces a function on the base space of the fibration. More information can be found in this answer: mathoverflow.net/questions/20740/… – Dmitri Pavlov Sep 20 '11 at 12:48