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, e.g. 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 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 measure space, such measures are not always probability measures. In general, in statistical physics, if we consider sentences of the form "the probability of a system S assuming state A is p" the geometry of the system does not always lead to the definition of a probability measure under congruence, although it may do so in the case of systems with just one degree of freedom.
Probability measures are also used in mathematical biology. For instance, in comparative sequence analysis a probability measure may be defined for the likelihood that a variant may be permissible for an amino acid in a sequence.
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