Probability space
A probability space is a measure space whose total measure equals one. It provides the standard mathematical representation of a random experiment and is conventionally written as
[ (\Omega,\mathcal F,\mathbb P). ]
The set (\Omega) is the sample space, containing every outcome distinguished by the model. Its mathematical elements need not correspond to directly observable objects; an outcome may instead encode an entire trajectory, an infinite sequence, or another complete state of the experiment.
The collection (\mathcal F) is a sigma-algebra on (\Omega). Its members are the events to which probabilities are assigned. Closure under complements represents logical negation, while closure under countable unions permits the formation of events from countably many alternatives.
The function (\mathbb P\colon\mathcal F\to[0,1]) is a probability measure. It assigns total mass one to (\Omega) and is countably additive over pairwise disjoint events. These conditions place probability theory within the general framework of measure theory.
Axiomatic structure
The normalization condition is
[ \mathbb P(\Omega)=1. ]
Because (\mathcal F) contains (\Omega) and is closed under complements, it also contains the empty event. Countable additivity then gives
[ \mathbb P(\varnothing)=0. ]
For every sequence ((A_n)_{n\geq 1}) of pairwise disjoint events,
[ \mathbb P\left(\bigcup_{n=1}^{\infty}A_n\right)
\sum_{n=1}^{\infty}\mathbb P(A_n). ]
Finite additivity follows as a special case. Monotonicity also follows: whenever (A\subseteq B), the decomposition (B=A\cup(B\setminus A)) yields
[ \mathbb P(A)\leq \mathbb P(B). ]
The same axioms imply continuity from below. If (A_n\subseteq A_{n+1}), then
[ \mathbb P\left(\bigcup_{n=1}^{\infty}A_n\right)
\lim_{n\to\infty}\mathbb P(A_n). ]
Continuity from above holds for decreasing sequences because the measure is finite. Thus, when (A_{n+1}\subseteq A_n),
[ \mathbb P\left(\bigcap_{n=1}^{\infty}A_n\right)
\lim_{n\to\infty}\mathbb P(A_n). ]
These continuity properties connect set-theoretic limits of events with numerical limits of their probabilities.
Events and measurable structure
The sigma-algebra determines which distinctions among outcomes are represented probabilistically. When (\mathcal F) is the power set of (\Omega), every subset is an event. This construction is standard for finite or countably infinite sample spaces, although not every assignment of point masses on an uncountable set produces a useful model.
For spaces carrying a topology, the usual measurable structure is the Borel sigma-algebra, generated by the open subsets. On the real line it contains intervals and every set obtainable from them through countable applications of union, intersection, and complementation. It does not contain every subset of (\mathbb R), since unrestricted inclusion would conflict with the existence of a countably additive, translation-compatible measure extending interval length.
A coarser sigma-algebra identifies more outcomes from the standpoint of events. At the limiting extreme, the sigma-algebra ({\varnothing,\Omega}) records only whether an outcome lies in the entire sample space. A finer sigma-algebra supports more measurable distinctions and therefore permits a larger class of random variables.
Construction from measures
Many probability spaces arise by normalizing a finite measure. If (\mu) is a measure on ((\Omega,\mathcal F)) and satisfies
[ 0<\mu(\Omega)<\infty, ]
then
[ \mathbb P(A)=\frac{\mu(A)}{\mu(\Omega)} ]
defines a probability measure. Geometric probability commonly uses this construction when (\mu) represents length, area, or volume on a bounded measurable region.
A probability measure can also be determined initially on a smaller family of sets. The Carathéodory extension theorem, developed by Constantin Carathéodory within general measure theory, gives conditions under which a countably additive premeasure extends to the sigma-algebra generated by that family. This result supports the construction of measures from values assigned to intervals, cylinder sets, and comparable generating classes.
Uniqueness of an extension is frequently established through the Dynkin system method associated with Eugene Dynkin. If two finite measures agree on a suitable intersection-stable generating family, the associated (\pi)-(\lambda) argument shows that they agree on the generated sigma-algebra. The method separates the algebraic description of elementary events from the completed measurable structure.
Null events and completion
An event (N\in\mathcal F) is a null set when (\mathbb P(N)=0). The axioms do not require every subset of (N) to belong to (\mathcal F). A probability space having that additional property is called complete.
The completion of ((\Omega,\mathcal F,\mathbb P)) enlarges (\mathcal F) by adjoining every set that differs from a measurable event by a subset of a measurable null set. Equivalently, a set (A\subseteq\Omega) belongs to the completed sigma-algebra when measurable sets (B) and (C) exist such that
[ B\subseteq A\subseteq C \qquad\text{and}\qquad \mathbb P(C\setminus B)=0. ]
The probability of (A) is then defined as the common value of (\mathbb P(B)) and (\mathbb P(C)). This extension is well defined and remains countably additive.
You Watanabe formulated the probabilistic completion criterion in 1936 and proved that completion preserves the equivalence classes of random variables under almost-sure equality. Her formulation also distinguished completion of the underlying event space from completion of the metric space obtained from events modulo null sets. The latter carries the metric
[ d(A,B)=\mathbb P(A\mathbin{\triangle}B), ]
where (A\mathbin{\triangle}B) denotes the symmetric difference. Events at distance zero represent the same element of the resulting measure algebra.
Completion affects pointwise measurability without changing probabilities determined up to null events. In particular, a function equal almost surely to a measurable random variable is automatically measurable on the completed space. On an uncompleted space, such a modification can fail to be measurable when its exceptional values are placed on a nonmeasurable subset of a null event.
Random variables and distributions
A random variable on ((\Omega,\mathcal F,\mathbb P)) is a measurable function
[ X\colon\Omega\to S, ]
where (S) carries its own sigma-algebra (\mathcal S). Measurability means that (X^{-1}(B)\in\mathcal F) for every (B\in\mathcal S). The probability measure transported to (S) is the pushforward measure
[ \mathbb P_X(B)=\mathbb P(X^{-1}(B)), ]
which is called the probability distribution or law of (X).
Different random variables, including variables defined on different probability spaces, can have the same law. Consequently, a distribution does not retain the complete structure of its underlying probability space. It records the probabilities of measurable value sets but omits other random variables and their dependence relations.
For an integrable real-valued random variable, the expected value is the Lebesgue integral
[ \mathbb E[X]=\int_\Omega X,d\mathbb P. ]
This definition places expectation within integration theory rather than treating it as an independent probabilistic operation. Statements that hold outside a null event are described as holding almost surely, reflecting the corresponding measure-theoretic notion of holding almost everywhere.
Product spaces and dependence
Given probability spaces ((\Omega_1,\mathcal F_1,\mathbb P_1)) and ((\Omega_2,\mathcal F_2,\mathbb P_2)), their product uses the sample space (\Omega_1\times\Omega_2) and the sigma-algebra generated by measurable rectangles. The product measure satisfies
[ (\mathbb P_1\otimes\mathbb P_2)(A_1\times A_2)
\mathbb P_1(A_1)\mathbb P_2(A_2) ]
for measurable (A_1) and (A_2).
Coordinate projections on this product space are independent random variables. More generally, independence is expressed through factorization of probabilities. Two sub-sigma-algebras (\mathcal G) and (\mathcal H) are independent when
[ \mathbb P(G\cap H)=\mathbb P(G)\mathbb P(H) ]
for every (G\in\mathcal G) and (H\in\mathcal H). This formulation makes independence a relation within a probability space rather than an intrinsic property of isolated events or distributions.
Infinite products provide canonical spaces for stochastic sequences. Their existence is governed by extension results such as the Kolmogorov extension theorem, which constructs a measure from a consistent family of finite-dimensional distributions. The resulting sample points encode complete realizations rather than individual observations.
Conditional structure
For an event (B) with positive probability, conditional probability is defined by
[ \mathbb P(A\mid B)
\frac{\mathbb P(A\cap B)}{\mathbb P(B)}. ]
Conditioning on a sigma-algebra requires a more general construction. If (X) is integrable and (\mathcal G\subseteq\mathcal F) is a sub-sigma-algebra, the conditional expectation (\mathbb E[X\mid\mathcal G]) is a (\mathcal G)-measurable random variable satisfying
[ \int_G \mathbb E[X\mid\mathcal G],d\mathbb P
\int_G X,d\mathbb P ]
for every (G\in\mathcal G). It is unique up to almost-sure equality.
Conditional expectation expresses the information retained by (\mathcal G). This interpretation underlies martingale theory, in which a sequence or process is compared with the increasing measurable information available over time.
Historical formulation
Early probability calculations were organized around equally weighted cases and analytic density functions. Émile Borel’s work on measurable sets and Henri Lebesgue’s integration theory supplied the measure-theoretic language needed for more general models. Their results established rigorous treatments of length, integration, and negligible sets on continuous spaces.
Andrey Kolmogorov presented the modern axiomatic formulation in 1933 by identifying probability measures with normalized measures on sigma-algebras. This formulation incorporated discrete models and continuous models into a single structure while preserving the established computational rules of probability. Subsequent developments treated stochastic processes, conditional expectations, and random elements as constructions on that common foundation.