Probability measure

A probability measure assigns numbers from zero to one to events, with total probability one and countable additivity across disjoint events.

Sample space: The set of all possible outcomes of a random experiment. It contains the outcomes on which the measure's event probabilities are defined.
Probability space: A triple consisting of a sample space, a sigma-algebra, and a probability measure. It packages the measure with the outcomes and events on which it is defined.
Probability distribution: A mathematical description of the probabilities assigned to possible values of a random quantity. A random variable's distribution is the probability measure it induces on its value space.
Probability density function: A function whose integral over a region gives that region's probability for a continuous distribution. A density represents a measure relative to a reference measure but is not itself event probability.
Event (probability theory): A measurable subset of a sample space to which a probability can be assigned. Events are the sets that a probability measure evaluates.
Pushforward measure: A measure transferred through a measurable function to assign mass to its output values. Random variables induce output distributions by pushing a probability measure forward.
Expected value: The probability-weighted average of a random variable's values, defined by integration. It summarizes a random quantity by integrating it against its probability measure.
Probability mass function: A function assigning probabilities to individual values of a discrete random variable. For discrete spaces, point probabilities can represent the measure directly.
Sigma-algebra: A collection of subsets closed under complements and countable unions, including the empty set. It specifies which subsets of the sample space count as measurable events.
Product measure: A measure on a product space whose values on measurable rectangles multiply the factor measures. It constructs joint probability models from measures on separate spaces.

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