What is E[|X|] ?

|X| is the absolute value of X, i.e. √X^2.

 

Yes, it is the expectation value of the absolute value of x. E.g. if X has a uniform distribution in -1<=x<-1 then E(X)=0 but |X| has a uniform distribution 0<=|x|<=1, so that E(|X|)=0.5.
No idea about the double bar in Martingales. Usually, this nomenclature is explained at the beginning of the article or book where it is used.

 

I simply don't believe that a book on probability would use "E(x)" without explaining that it means the expected value, and explaining with at expected value meant. I can believe that it would not define the absolute value because you should have learned that in a basic algebra class. And your question, as stated, doesn't really have anything to do with "martingales".

 

I also suppose that || means conditional expectation in that case, especially as it coincides with the definition in Wikipedia. However if you have doubt you have to look through the book where this nomenclature first appears. I did not say that it necessarily appears in the list of symbols. Most books lack a list of symbols at all.

 

I don't have much knowledge of Martingales but the 4th condition guarantees that the distribution of each subsequent random variable, X_{n+1} is centered around X_n in F_n.

As an example, let our random variable X_n represent the position after n steps in a one-dimensional random walk. The fourth condition imposes that it is equally likely to step forward (to X_n + 1) as it is to step backwards (to X_n - 1). It's really just a conditional probability problem since the position after n+1 steps depends on the position after n steps.