is probability defined on outcomes or events (or both)? studying myself and trying to understand probability foundations, and I find that even books do not agree, or probably I do not understand.
What I have learned:
Outcome: what can happen in an experiment.  I think these are exclusive, for any experiment only one outcome not several. I think it also called "elementary event". Sample space is the set of all outcome.
      Example: throw the die, the outcome is 1 or 2 or ... or 6.
Event: a grouping or function(?) on the outcome. Example: the outcome is even.
Random Variable: a mapping from ___ to real numbers.
Probability: a mapping from ___ to probability, that acts like a measure (additive, ...).
The question is about the empty parts ___ above, and specifically for probability (since we can only ask one question each post). 
Is probability a map from outcomes, or events?
I have looked in different books and get more confused.
Book1 suggests is "outcomes": "All elementary events for a sample space. The probability of a random event is a measure on this set."
Book2: "probabilities are assigned to events"
 A: Outcomes are indeed a word used for the "elementary events". So they are only a subset of the possible "events" to which we can assign probabilities.
Standard example: let a six-size fair die thrown once. The "outcomes/elementary events" are $(1,2,3,4,5,6)$.
But an "event" is also, "the die came up even". This is not an elementary event, but we want and can assign probabilities also to it.
Another non-elementary event is "The value of the die is less than 3".  Etc.
So probabilities are assigned to events, that include also the elementary events.
A: A probability is a mapping from the collection of events to [0,1]. Eg, for a dice the event "even outcome" is 1/2. The event "1" takes probability 1/6. The collection of events consists of all subsets of outcomes (called the power set).
A random variable is a mapping from some background space to the real numbers. Eg, for a dice it takes the numbers 1,2,3,4,5,6.
A: the simplified, mathematical ituition of a random variable is that it is a function from a measure space into the real numbers. 
A measure space is a space with a measure. 
And measure is necessary to quantify the probability of the events (and event is a possible outcome of an experiment) in that space. 
Example: You flip a coin. Measure Space has two events: Head and tails and it holds the probabilities of those events: eg. 0.4 and 0.6
Now, you dont want a space with heads and tails or other weird things. Therefore the mapping from this weird space with the measure into the real numbers. 
Plus: the random variable transmits the information about the event probabilities into the real numbers.
So that flipping a coin becomes a bernoulli experiment with 0-1 outcomes   
