1.4.3 Probability Measure
A probability measure is a function that assigns a probability to each event within the event space, the probability is a real number between 0 and 1. This can be denoted as:
As 
is a function, the input to this function is an event and the output is
the probability of that event. A completely impossible event has a probability of 0, whereas an event that
is certain to occur has a probability of 1. However, apart from the
assignment of 1 to the sample space set and 0 to the empty set, the
probability measure function doesn't provide specific guidance or rules for
determining the initial values (probabilities) that should be assigned to
different events. For guidance with that, we need to turn to the
interpretations of probability, like classical, empirical, or subjective
(discussed earlier in section 2).
To understand it with an analogy: Think of the probability measure as a “black box” into which you input an event, and it outputs a number between 0 and 1. However, this definition doesn’t tell you the internal workings of this “black box” for any specific random experiment.
For instance, in the context of rolling a fair six-sided die:
- The probability measure tells you that the probability of getting any one specific…