 in kolmogorov's probability theory, the probability p of some event e, denoted p(e), is usually defined such that p satisfies the kolmogorov axioms, named after the famous russian mathematician andrey kolmogorov, which are described below. these assumptions can be summarised as: let (Î©, f, p) be a measure space with p(Î©)=1. then (Î©, f, p) is a probability space, with sample space Î©, event space f and probability measure p. an alternative approach to formalising probability, favoured by some bayesians, is given by cox's theorem. == axioms == === first axiom === the probability of an event is a non-negative real number: :p(e)\in\mathbb{r}, p(e)\geq 0 \qquad \forall e\in f where f is the event space. in particular, p(e) is always finite, in contrast with more general measure theory. theories which assign negative probability relax the first axiom. === second axiom ===  this is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. more specifically, there are no elementary events outside the sample space. : p(\omega) = 1. this is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either. ===third axiom=== this is the assumption of &sigma;-additivity: : any countable sequence of disjoint (synonymous with mutually exclusive) events e_1, e_2, ... satisfies ::p(e_1 \cup e_2 \cup \cdots) = \sum_{i=1}^\infty p(e_i). some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a &sigma;-algebra. quasiprobability distributions in general relax the third axiom. ==consequences== from the kolmogorov axioms, one can deduce other useful rules for calculating probabilities. ===the probability of the empty set=== : p(\emptyset)=0. ===monotonicity=== : \quad\text{if}\quad a\subseteq b\quad\text{then}\quad p(a)\leq p(b). ===the numeric bound=== it immediately follows from the monotonicity property that : 0\leq p(e)\leq 1\qquad \forall e\in f. ==proofs== the proofs of these properties are both interesting and insightful. they illustrate the power of the third axiom, and its interaction with the remaining two axioms. when studying axiomatic probability theory, many deep consequences follow from merely these three axioms. in order to verify the monotonicity property, we set e_1=a and e_2=b\backslash a, where \quad a\subseteq b \text{ and } e_i=\emptyset for i\geq 3. it is easy to see that the sets e_i are pairwise disjoint and e_1\cup e_2\cup\ldots=b. hence, we obtain from the third axiom that : p(a)+p(b\backslash a)+\sum_{i=3}^\infty p(\emptyset)=p(
Mathematical_axioms Probability_theory