Probability space

The probability space is a fundamental concept from the mathematical branch of probability theory. It is a mathematical model for describing random experiments. Here, the various possible outcomes of the experiment are combined into a set. Conscience subsets of this result set can then be assigned numbers between 0 and 1, which are interpreted as probabilities.

The concept of probability space was introduced in the 1930s by the Russian mathematician Andrei Kolmogorov, the order axiomatization of probability theory succeeded (see also: Kolmogorov axioms ).

Definition

A probability space is a measure space whose measure is a probability measure.

In detail, this means:

• Is an arbitrary non-empty set, called the result set. Its elements are called outcomes.
• Is a σ - algebra over the base set, so a lot is composed of subsets of containing and closed under the formation of complements and countable unions. The elements of are called events. The σ - algebra itself is also called algebra of events or event field.
• Is a measure, ie a function that assigns numbers to the events, so that applies and for pairwise disjoint events.

The measurement space is also called the sample space. A probability space is thus an event space on which a probability measure is given in addition.