Why not just assign probabilities to every subset of the sample space?

For uncountable sample spaces no consistent countably additive probability can be defined on all subsets, so probabilities are restricted to a sigma-algebra of measurable events, which still contains every event of practical interest.

What does almost surely mean?

An event holds almost surely if its complement has probability zero; such null events can be ignored for the purpose of computing probabilities and expectations even though they are not literally impossible.

Probability Spaces and Events

A probability space is the triple consisting of a sample space of outcomes, a sigma-algebra of events, and a probability measure assigning each event a number between zero and one, and it is the stage on which all of probability theory is set.

Definition

A probability space is a triple consisting of a sample space, a sigma-algebra of measurable subsets called events, and a countably additive probability measure of total mass one that assigns each event its probability.

Scope

The topic covers the sample space and the sigma-algebra of events, the axioms a probability measure must satisfy, continuity of probability along increasing and decreasing sequences of events, the construction of measures from set functions via Caratheodory extension, and standard constructions such as Lebesgue measure on the unit interval as a canonical probability space.

Core questions

What is the difference between an outcome and an event, and why must events form a sigma-algebra?
Which properties define a probability measure, and how do they yield continuity from below and above?
How is a probability measure constructed from a description of probabilities on simple sets?
What canonical probability space underlies familiar models such as a uniform random number on the unit interval?

Key concepts

sample space and outcomes
sigma-algebra of events
countable additivity
continuity of probability
null events and almost-sure properties

Key theories

Axioms of a probability measure: A probability measure is non-negative, assigns the whole sample space probability one, and is countably additive over disjoint events; these axioms imply monotonicity, the inclusion-exclusion formula, and continuity along monotone sequences of events.
Caratheodory extension theorem: A countably additive set function defined on an algebra extends uniquely to a measure on the generated sigma-algebra, which is what allows a probability measure to be specified on simple events and then extended to all measurable events.

Clinical relevance

The probability-space formalism is what makes statements about random phenomena unambiguous; every applied probabilistic model, from queueing systems to statistical inference and risk modeling, is implicitly an assertion about a probability space and the events defined on it.

History

Although informal probabilities were computed for centuries, the precise notion of a probability space dates to Kolmogorov's 1933 axiomatization, which borrowed Caratheodory's extension machinery from measure theory to give events and their probabilities a rigorous home.

Key figures

Andrey Kolmogorov
Constantin Caratheodory
Emile Borel

Seminal works

kolmogorov1933

Frequently asked questions

Why not just assign probabilities to every subset of the sample space?: For uncountable sample spaces no consistent countably additive probability can be defined on all subsets, so probabilities are restricted to a sigma-algebra of measurable events, which still contains every event of practical interest.
What does almost surely mean?: An event holds almost surely if its complement has probability zero; such null events can be ignored for the purpose of computing probabilities and expectations even though they are not literally impossible.