Probability 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. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probability levels are also the same.

A probability space consists of three parts:

  1. A sample space, Ω, which is the set of all possible outcomes.
  2. A set of events, where each event is a set containing zero or more outcomes.
  3. The assignment of probabilities to the events, that is, a function P from events to probability levels.

An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex events are used to characterize groups of outcomes. The collection of all such events is a σ-algebra . Finally, there is a need to specify each event's likelihood of happening. This is done using the probability measure function, P.

Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ω, from the sample space Ω. All the events in that contain the selected outcome ω (recall that each event is a subset of Ω) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function P.

The prominent Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. Nowadays alternative approaches for axiomatization of probability theory exist; see “Algebra of random variables”, for example.

This article is concerned with the mathematics of manipulating probabilities. The article probability interpretations outlines several alternative views of what "probability" means and how it should be interpreted. 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.

Read more about Probability Space:  Introduction, Definition, Discrete Case, General Case, Non-atomic Case, Complete Probability Space

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