For a die roll, the ‘Universe’ is simply {1,2,3,4,5,6}. At least, this is the normal way it is done in probability theory.
No, I’m reasonably confident that in the example the author meant ‘Universe’ to mean {all possible states of the universe after the dice role}. The random variable X is a function mapping from states of the universe to values at the top of the die. The inverse-image#Inverseimage) of {4} is the set of all states of the universe where the die landed on 4. That inverse-image defines an [event](http://en.wikipedia.org/wiki/Event(probability_theory)). The measure of that event is what we mean when we say ‘probability of the die landing on 4’.
And it still doesn’t answer what he means by ‘closed system’.
I’m not entirely sure either. Rather than take a guess, I’ll let twanvl speak for him/herself.
No, I’m reasonably confident that in the example the author meant ‘Universe’ to mean {all possible states of the universe after the dice role}. The random variable X is a function mapping from states of the universe to values at the top of the die. The inverse-image#Inverseimage) of {4} is the set of all states of the universe where the die landed on 4. That inverse-image defines an [event](http://en.wikipedia.org/wiki/Event(probability_theory)). The measure of that event is what we mean when we say ‘probability of the die landing on 4’.
I’m not entirely sure either. Rather than take a guess, I’ll let twanvl speak for him/herself.