TITLE: Intuition behind Conditional Expectation
QUESTION [123 upvotes]: I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know. 
Let me get more specific. Let $\left(\Omega,\mathcal{A},P\right)$ be a probability space and $X$ an integrable real random variable defined on $(\Omega,\mathcal{A},P)$. Let $\mathcal{F}$ be a sub-$\sigma$-algebra of $\mathcal{A}$. Then $E[X|\mathcal{F}]$ is the a.s. unique random variable $Y$ such that $Y$ is $\mathcal{F}$-measurable and for any $A\in\mathcal{F}$, $E\left[X1_A\right]=E\left[Y1_A\right]$.
The common interpretation seems to be: "$E[X|\mathcal{F}]$ is the expectation of $X$ given the information of $\mathcal{F}$." I'm finding it hard to get any meaning from this sentence.

In elementary probability theory, expectation is a real number. So the sentence above makes me think of a real number instead of a random variable. This is reinforced by $E[X|\mathcal{F}]$ sometimes being called "conditional expected value". Is there some canonical way of getting real numbers out of $E[X|\mathcal{F}]$ that can be interpreted as elementary expected values of something?
In what way does $\mathcal{F}$ provide information? To know that some event occurred, is something I would call information, and I have a clear picture of conditional expectation in this case. To me $\mathcal{F}$ is not a piece of information, but rather a "complete" set of pieces of information one could possibly acquire in some way. 

Maybe you say there is no real intuition behind this, $E[X|\mathcal{F}]$ is just what the definition says it is. But then, how does one see that a martingale is a model of a fair game? Surely, there must be some intuition behind that!
I hope you have got some impression of my misconceptions and can rectify them.

REPLY [3 votes]: I was only able to understand the notion of the conditional expectation with respect to a sub-$σ$-algebra $\mathcal F$, when I realized that this game is only interesting when $\mathcal F$ is "not Hausdorff", meaning that there might be points $x$ and $y$ which cannot be separated by an $\mathcal F$-measurable set.  Any $\mathcal F$-measurable function must therefore coincide on $x$ and $y$, so $E(X|\mathcal F)$ tries to be the best photograph of the random variable $X$ which coincides on $x$ and $y$, as well as on any other similar pairs of points.
In the event that $\mathcal F$ is the smallest sub-$σ$-algebra possible, namely  $\mathcal F = \{\emptyset,  \Omega\}$,
only constant functions are measurable.   So $E(X|\mathcal F)$ must be a constant function, and that constant  turns out to be the
average of $X$, a.k.a. the expectation of $X$.

PS.  This is a comment I made in a recent question (Refference for conditional expectation) which in turn brought me here to this 10 year old question when I clicked on a "Related" post.  Reading the answers I did not find anyone referring to the above point of view, so I hope my little contribution will be useful to someone.