Probability space

A probability space is a measure space consisting of a sample space, a σ-algebra of events, and a probability measure that assigns each event a value from zero to one.

Sample space: The set of all possible outcomes of a random experiment. It supplies the outcomes that the probability space organizes.
Kolmogorov axioms: The axioms requiring nonnegative probabilities, total probability one, and countable additivity. They characterize which assignments make the measure a probability measure.
Random variable: A measurable function from a probability space to a measurable value space. It converts outcomes into numerical or structured quantities while preserving measurable events.
Probability density function: A function whose integral over a region gives that region's probability for a continuous random variable. A density represents probabilities in a chosen coordinate space, rather than defining the full abstract space.
Sigma-algebra: A collection of sets closed under complements and countable unions, including the empty set and the whole space. It determines which subsets of outcomes qualify as events.
Countable additivity: The property that a measure of a countable disjoint union equals the sum of the measures. It governs probabilities of countably many mutually exclusive events.
Probability distribution: A probability measure describing the possible values of a random variable and their probabilities. It is induced when the space's measure is transferred through a random variable.
Lebesgue measure: A measure on Euclidean spaces that generalizes length, area, and volume. It need not have total measure one, unlike the measure of a probability space.
Probability measure: A countably additive measure whose total value on the whole space is one. It assigns probabilities to the events in the sigma-algebra.
Almost surely: A property holds almost surely when the outcomes where it fails form a probability-zero event. The measure distinguishes certainty in probability from certainty at every outcome.

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