In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function to be a probability measure on a probability space are that: must return results in the unit interval returning for the empty set and for the entire space. must satisfy the countable additivity property that for all countable collections of pairwise disjoint sets: For example, given three elements 1, 2 and 3 with probabilities and the value assigned to is as in the diagram on the right. The conditional probability based on the intersection of events defined as: satisfies the probability measure requirements so long as is not zero. Probability measures are distinct from the more general notion of fuzzy measures in which there is no requirement that the fuzzy values sum up to and the additive property is replaced by an order relation based on set inclusion. Example applications Market measures which assign probabilities to financial market spaces based on actual market movements are examples of probability measures which are of interest in mathematical finance; for example, in the pricing of financial derivatives. For instance, a risk-neutral measure is a probability measure which assumes that the current value of assets is the expected value of the future payoff taken with respect to that same risk neutral measure (i.e. calculated using the corresponding risk neutral density function), and discounted at the risk-free rate. If there is a unique probability measure that must be used to price assets in a market, then the market is called a complete market. Not all measures that intuitively represent chance or likelihood are probability measures. For instance, although the fundamental concept of a system in statistical mechanics is a