Editing Probability theory (section)

===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 <math>(\delta[x] + \varphi(x))/2</math>, where <math>\delta[x]</math> 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 <math>\Omega\,</math> (also called {{em|sample space}}) and a [[sigma-algebra|σ-algebra]] <math>\mathcal{F}\,</math> on it, a [[measure (mathematics)|measure]] <math>P\,</math> defined on <math>\mathcal{F}\,</math> is called a {{em|probability measure}} if <math>P(\Omega)=1.\,</math>

If <math>\mathcal{F}\,</math> is the [[Borel algebra|Borel σ-algebra]] on the set of real numbers, then there is a unique probability measure on <math>\mathcal{F}\,</math> for any CDF, and vice versa. The measure corresponding to a CDF is said to be {{em|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 <math>E\,</math> in the σ-algebra <math>\mathcal{F}\,</math> is defined as
<!--the correct formulation; X has nothing to do with it-->
:<math>P(E) = \int_{\omega\in E} \mu_F(d\omega)\,</math>
where the integration is with respect to the measure <math>\mu_F\,</math> induced by <math>F\,.</math>

Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside <math>\mathbb{R}^n</math>, as in the theory of [[stochastic process]]es. 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.