This is probably a dumb question but I can't wrap my head around it. Why is the formal definition of a random variable this way? What is the use of the little y variable in this defintion? Why does Y(?) got to be less or equal to y , won't it make more sense to be defined like {?|Y(?)}belongs to F. Furthermore what is the use of saying every elementary outcome ? belongs in F field, doesn't every ? belong to the F?