I've been learning all about conditional expectation these past couple of days and I am still having problems figuring out the principle idea behind the following expression, which constitutes the second fundamental property of the random variable E[X|Y]: $$ \int_{Y^{-1}(B)} E[X|Y]\, dP =\int_{Y^{-1}(B)} X\, dP$$
I just don't see the intuition as, for instance, requiring each "concept" of distance to obey the property of symmetry.
I made an example where X represents height, Y age and thus $\omega \mapsto X(\omega) $ would represent the actual height of the person, whose characteristics are encoded in $\omega$ and $\omega \mapsto E[X|Y](\omega) $ would assign each $\omega$ the expected value of the age group that the individual is in (so E[X|Y] is a function of Y, the first fundamental property of E[X|Y]). So, given this, what does the abovementioned property trying to say?
Thank you in advance!!
I hope this isn't just semantics and that I'm reading you're question correctly, but the expression you write is less of a "fundamental property" of $\mathbb{E}[X\vert Y]$ than it is a definition.
Intuitively, $\mathbb{E}[X\vert Y]$ denotes the expectation of X conditional on the "$\sigma$-algebra" generated by the random variable $Y$ (which we denote $\sigma(Y)$). If you're unfamiliar with what a $\sigma$-algebra is, you can think of it as the collection of sets for which the random variable, $Y$, is measurable. i.e. if we observe that that $Y$ takes some set of values in $\mathbb{R}$, then we can theoretically determine the probability that generated the values of $Y(\omega)$ that we observe.
The sets which make up $\sigma(Y)$ are exactly the sets $Y^{-1}(B)\in \Omega$, where $B$ is some arbitrary Borel-set in $\mathbb{R}$.
By definition, $\mathbb{E}[X\vert Y]$ is the $\sigma(Y)$-measurable random variable that satisfies the expression
$$\int_{Y^{-1}(B)}\mathbb{E}[X\vert Y] \,d\mathbb{P} = \int_{Y^{-1}(B)} X \,d\mathbb{P} \quad \star $$
for any Borel set $B$.
The reason why we care about this expression/definition is because we can prove it exists and (more importantly) is uniquely determined by the Radon-Nikodym Theorem. So if we can prove some other random variable $Z$ satisfies $\star$, then we know that $Z = \mathbb{E}[X\vert Y]$ almost everywhere.
As far as a more intuitive interpretation. Consider a sample of 100 people, 50 men and 50 women. Let $X =$ height of any person in the sample and $Y = 1$ if male and 0 if female.
Then clearly, $\mathbb{E}[X\vert Y]$ is the expected height of a person in the sample, conditional on the person being male.
Then we can see that all $\star$ is really saying "if we restrict the sample of 100 people to include ONLY the 50 men, what is the expected height of a randomly drawn person from this restricted sample, conditional on that person being a male?"
Well, its obviously equal to the expected value of the height of any person we draw from this restricted sample, since they are all males.