probability space

In mathematics, a probability space is a fundamental concept used to model random phenomena. It consists of three components: a sample space (the set of all possible outcomes), a set of events (subsets of the sample space), and a probability measure that assigns a probability to each event.12

The sample space can be finite, like the outcomes of rolling a die, or infinite, such as the number of times a coin is tossed until heads appears. Events are specific outcomes or collections of outcomes we are interested in. The probability measure assigns a numerical value between 0 and 1 to each event, indicating its likelihood of occurring, with 1 representing certainty and 0 representing impossibility.21

Measure theory provides a rigorous framework for defining probability spaces, especially for infinite sample spaces. This approach, notably developed by Andrey Kolmogorov, allows for the consistent assignment of probabilities to events, forming the bedrock of modern probability theory and its applications in statistics and various scientific fields.13