Wahrscheinlichkeitsmass

Wahrscheinlichkeitsmass is a fundamental concept in probability theory and measure theory. It is a function that assigns a numerical value, representing a probability, to every event in a sample space. This value is always non-negative, and the probability of the entire sample space occurring is 1. Additionally, for any countable collection of disjoint events, the probability of their union is the sum of their individual probabilities. This property is known as countable additivity.

A probability measure is defined on a sigma-algebra, which is a collection of subsets of the sample space that are considered "measurable events." The sigma-algebra ensures that we can consistently assign probabilities to complex events. The triplet (Ω, F, P) is often used to denote a probability space, where Ω is the sample space, F is the sigma-algebra of events, and P is the probability measure.

The concept of a probability measure allows for a rigorous mathematical framework to describe randomness and uncertainty. It is essential for defining random variables, expectations, and various stochastic processes. The development of probability measures by mathematicians like Kolmogorov laid the foundation for modern probability theory and its applications in fields such as statistics, physics, finance, and computer science. Understanding probability measures is key to comprehending the behavior of random phenomena.