Probability Spaces

Parent: Probability

Definition

A probability space is a 3-tuple (\Omega, F, P) consisting of

1. A set \(\Omega\) called the sample space i.e. the set of all possible outcomes
2. A \(\sigma\)-algebra \(F\) on the set \(\Omega\), which informally specifies the set of all “possible” events (events are outcomes or combinations of outcomes)
3. A probability measure \(P: F \rightarrow [0, 1]\)

The 2-tuple \((\Omega, F)\) is called a measure space.

Related Material

A measurable space (also called a Borel space) is a 2-tuple consisting of

1. A set \(X\)
2. A A \(\sigma\)-algebra on the set \(X\)

When the 2-tuple measure space is equipped with a measure \(\mu\), becoming a 3-tuple, the measurable space becomes a measure space.