Probability Theory 2 | Probability Measures [dark version]

Probability measures formalize randomness by assigning probabilities to “events” in a way that behaves like area: the total probability of all outcomes is 1, impossible events have probability 0, and probabilities add up consistently for disjoint events. The core setup starts with a sample space Ω, the set of all possible outcomes of a random experiment, and then chooses a collection of subsets of Ω—called a σ-algebra A—so that the probability function can be defined without contradictions.

A probability measure P is a map from events (elements of A) to numbers in the interval [0,1]. Two boundary conditions anchor the definition: P(Ω)=1 and P(∅)=0. The additivity rule then handles how probabilities combine. If A and B are disjoint (their intersection is empty), then P(A∪B)=P(A)+P(B). The definition extends beyond two sets: for countably many pairwise disjoint events {A_i}, the probability of their union equals the sum of their probabilities. This “countable additivity” is what makes the framework work not just for finite partitions, but also for limits of increasingly fine event collections.

To make these rules meaningful, the domain of P must be restricted to a σ-algebra A. A σ-algebra is a family of subsets of Ω that is closed under three operations: it contains the empty set and the whole space (implicitly via the usual σ-algebra requirements), it is closed under complements (if A is an event, then Ω\A is also an event), and it is closed under countable unions (if A_1, A_2, … are events, then their union is also an event). In probability language, the elements of a σ-algebra are exactly the events whose probabilities are allowed to be defined.

The transcript illustrates the machinery with a single die. The sample space Ω is {1,2,3,4,5,6}. For a finite sample space, a natural choice is to take the σ-algebra A as the power set of Ω, meaning every subset is an event. With the usual “fair die” model, each outcome has probability 1/6, so P({2})=1/6. The event “even number” corresponds to {2,4,6}, giving P(even)=3/6=1/2. This example shows how probability measures turn everyday counting into a consistent rule system.

A closing exercise ties the definition together: using P(Ω)=1 and the additivity/complement structure, one can prove that for any event A, the probability of its complement satisfies P(A^c)=1−P(A). That identity is presented as a direct consequence of the probability measure axioms and becomes a recurring tool in later examples.