Kansmaat

Kansmaat, literally “probability measure” in Dutch mathematics, is the fundamental object in probability theory. It is a function μ defined on a σ-algebra F of subsets of a set Ω (the sample space) with values in the interval [0,1], such that μ(Ω) = 1 and, for any countable collection of pairwise disjoint sets A1, A2, … in F, μ(∪i Ai) = Σi μ(Ai). This property is Kolmogorov’s additivity axiom and underlies the way probabilities assign to events.

A probability space is the triple (Ω, F, μ). For any event A ∈ F, μ(A) is interpreted as the probability of A. The kansmaat is monotone: if A ⊆ B, then μ(A) ≤ μ(B). It is also continuous from below and above, expressing limit behavior of probabilities for increasing or decreasing sequences of events.

Examples help illustrate the concept. A discrete example: Ω = {H, T} for a coin toss, F = power set of Ω, and μ({H}) = p, μ({T}) = 1 − p, with 0 ≤ p ≤ 1. A continuous example: Ω = ℝ with the Borel σ-algebra and μ corresponding to a common distribution, such as the uniform distribution on [0,1], where μ([a,b]) = b − a for 0 ≤ a ≤ b ≤ 1.

Random variables are μ-integrable functions X: Ω → ℝ, with the expectation E[X] = ∫ X dμ. The kansmaat framework supports concepts such as independence, conditional probability, and the law of large numbers, making it central to probability theory, statistics, and related fields.