Math 2.0

definition

Probability measure

Definition

Let Ω\Omega be a set and let F\mathcal{F} be a σ-algebra on Ω\Omega. A probability measure on (Ω,F)(\Omega, \mathcal{F}) is a function P ⁣:F→[0,1]P \colon \mathcal{F} \to [0, 1] with these two properties. P(Ω)=1P(\Omega) = 1. For each sequence A1,A2,…A_1, A_2, \dots of elements of F\mathcal{F} with Ai∩Aj=∅A_i \cap A_j = \varnothing for all i≠ji \ne j, P(⋃n=1∞An)=∑n=1∞P(An).P\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} P(A_n).

Remarks

These are the probability axioms. In other words, a probability measure is a measure of total mass 11. Countable additivity implies P(∅)=0P(\varnothing) = 0 and finite additivity: P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B) for disjoint AA and BB. On a finite set Ω\Omega with the power set as F\mathcal{F}, each probability measure has the form P(A)=∑ω∈Ap(ω)P(A) = \sum_{\omega \in A} p(\omega) with p(ω)≥0p(\omega) \ge 0 and ∑ω∈Ωp(ω)=1\sum_{\omega \in \Omega} p(\omega) = 1.

Explanations

L1 human

A probability measure is a rule that gives each event a number between 00 and 11. The whole sample space gets 11, and the number of a union of separate pieces is the sum of the numbers of the pieces. For a fair die, each face gets 1/61/6, and the event "even" gets 3/6=1/23/6 = 1/2.

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Sources

  1. Web https://en.wikipedia.org/wiki/Probability_measure, Definition

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