Probability theory
Measure-theoretic probability theory

The utility of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two.

An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a PDF of (δ + φ (x)) / 2 {\displaystyle (\delta +\varphi (x))/2}, where δ {\displaystyle \delta } is the Dirac delta function.

Other distributions may not even be a mix, for example, the Cantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using measure theory to define the probability space:

Given any set Ω {\displaystyle \Omega \,} (also called sample space) and a σ-algebra F {\displaystyle {\mathcal {F}}\,} on it, a measure P {\displaystyle P\,} defined on F {\displaystyle {\mathcal {F}}\,} is called a probability measure if P (Ω) = 1. {\displaystyle P(\Omega)=1.\,}

If F {\displaystyle {\mathcal {F}}\,} is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on F {\displaystyle {\mathcal {F}}\,} for any CDF, and vice versa. The measure corresponding to a CDF is said to be induced by the CDF. This measure coincides with the pmf for discrete variables and PDF for continuous variables, making the measure-theoretic approach free of fallacies.

The probability of a set E {\displaystyle E\,} in the σ-algebra F {\displaystyle {\mathcal {F}}\,} is defined as

where the integration is with respect to the measure μ F {\displaystyle \mu _{F}\,} induced by F. {\displaystyle F\,.}

Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside R n {\displaystyle \mathbb {R} ^{n}}, as in the theory of stochastic processes. For example, to study Brownian motion, probability is defined on a space of functions.

When it is convenient to work with a dominating measure, the Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure. Discrete densities are usually defined as this derivative with respect to a counting measure over the set of all possible outcomes. Densities for absolutely continuous distributions are usually defined as this derivative with respect to the Lebesgue measure. If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions.
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