Probability space: A mathematical construct to model a random experiment

A probability space is a mathematical construct that represents a random experiment or system in the world. It defines the possible outcomes, the possible collections of outcomes (events), and the probability that each of those collections of outcomes happen. Therefore, it is like an enumeration of everything that could happen, together with how feasible it is to happen.

Formally, a probability space is a tuple $(\mathrm{\Omega },\mathcal{F},P)$ such that:

• The sample space $\mathrm{\Omega }$ is the set of all possible outcomes.
• The sigma algebra or $\sigma$-algebra $\mathcal{F}$ is set of all possible events, i.e. set of outcomes that are grouped together.
• The measure $P$ is the probability that each of the combinations in the $\sigma$-algebra happen (formally a measure).

NOTE that sample space, measurable space and probability space are increasingly larger constructs:

Sample space < Measurable space < Probability space.