# Probability space

**probability measuresGaussian measureoutcomesprobability measureprobability of an eventstate space**

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly.wikipedia

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### Probability theory

**theory of probabilityprobabilityprobability theorist**

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly.

Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space.

### Space (mathematics)

**spacemathematical spacespaces**

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly.

While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.

### Probability measure

**measureprobability distributionlaw**

This is done using the probability measure function, P.

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity.

### Probability axioms

**axioms of probabilityKolmogorov axiomsaxioms**

The Russian mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s.

Then (Ω, F, P) is a probability space, with sample space Ω, event space F and probability measure P.

### Event (probability theory)

**eventeventsrandom event**

So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see Events in probability spaces, below).

### Sample space

**event spacespacerepresented by points**

A well-defined sample space is one of three basic elements in a probabilistic model (a probability space); the other two are a well-defined set of possible events (a sigma-algebra) and a probability assigned to each event (a probability measure function).

### Outcome (probability)

**outcomeoutcomesequally likely outcomes**

So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events.

### Almost surely

**almost alwaysalmost surezero probability**

Let be a probability space.

### Randomness

**randomchancerandomly**

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly.

Considering the two events independently, one might expect that the probability that the other child is female is ½ (50%), but by building a probability space illustrating all possible outcomes, one would notice that the probability is actually only ⅓ (33%).

### Probability

**probabilisticprobabilitieschance**

In Kolmogorov's formulation (see probability space), sets are interpreted as events and probability itself as a measure on a class of sets.

### Sigma-algebra

**σ-algebraσ-algebrasigma algebra**

The collection of all such events is a σ-algebra \mathcal{F}.

Suppose is a probability space.

### Probability mass function

**mass functionprobability massmass**

Probabilities can be ascribed to points of \Omega by the probability mass function such that.

Suppose that is a probability space

### Quantum probability

In addition, there have been attempts to construct theories for quantities that are notionally similar to probabilities but do not obey all their rules; see, for example, free probability, fuzzy logic, possibility theory, negative probability, and quantum probability.

In classical probability theory, information is summarized by the sigma-algebra F of events in a classical probability space (Ω, F,P).

### Standard probability space

**a bijective measurable and measure preserving map whose inverse is also measurable and measure preservingisomorphism modulo zeroLebesgue space**

This is not a one-to-one correspondence between {0,1} ∞ and [0,1] however: it is an isomorphism modulo zero, which allows for treating the two probability spaces as two forms of the same probability space.

In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940.

### Measure (mathematics)

**measuremeasure theorymeasurable**

In short, a probability space is a measure space such that the measure of the whole space is equal to one.

. A probability space is a measure space with a probability measure.

### Random variable

**random variablesrandom variationrandom**

A random variable X is a measurable function X: Ω → S from the sample space Ω to another measurable space S called the state space.

In that context, a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers.

### Elementary event

**elementaryelementary eventssample points**

Under the measure-theoretic definition of a probability space, the probability of an elementary event need not even be defined.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

Any probability distribution defines a probability measure.

A measurable function between a probability space and a measurable space is called a discrete random variable provided that its image is a countable set.

### Measurable function

**measurableBorel functionLebesgue measurable function**

A random variable X is a measurable function X: Ω → S from the sample space Ω to another measurable space S called the state space.

In probability theory, a measurable function on a probability space is known as a random variable.

### Measure space

**measuremeasure spacesprobability space**

One important example of a measure space is a probability space.

### Borel set

**Borel algebraBorelBorel σ-algebra**

Here Ω = [0,1], is the σ-algebra of Borel sets on Ω, and P is the Lebesgue measure on [0,1].

Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel algebra.

### Filtration (probability theory)

**filtrationfiltered probability spacefiltrations**

Let be a probability space and let I be an index set with a total order \leq (often \N, \R^+, or a subset of \mathbb R^+).

### Talagrand's concentration inequality

In probability theory, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces.

### Algebra of random variables

Nowadays alternative approaches for axiomatization of probability theory exist, e.g. algebra of random variables.

### Probability interpretations

**philosophy of probabilityinterpretation of probabilityinterpretations of probability**

The article "probability interpretations" outlines several alternative views of what “probability” means and how it should be interpreted.