What is a probability measure and what do its values mean?

"A probability measure is simply a function from the domain of events to the codomain of real numbers. We’ll normally use the letters “Pr” for our probability measure. In symbols, Pr : P(Ω) → R, since the set of all events is the power set of the sample space. There’s actually another constraint, though, which is that Pr’s values must be in the range 0 to 1, inclusive. So it’s more correct to write: Pr : P(Ω) → [0, 1]. The meaning of the probability measure is intuitive enough: it indicates how likely we think each event is to occur. In the baby example, if we say Pr({boy}) = .5, it means there’s a .5 probability, or a 50% chance, that a male child will be born. In the game example, if we say Pr(M) = .667, it means there’s a two-thirds chance of winning the right to go first. In all cases, a probability of 0 means “impossible to occur” and a probability of 1 means “absolutely certain to occur.” In colloquial English, we most often use percentages to talk about these things, but from now on we’ll use probabilities and values in the 0 to 1 range rather than percent chances."