Probability Measure

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A probability measure is a real-valued measure on events which assigns a nonnegative probability value to every set in a sigma-field (a collection of subsets of a sample space).

(a) [math] 0 \leq P(A) \leq 1 [/math] for all subsets [math]A \; \in \; \mathcal{F}[/math]
(b) [math]P(\emptyset)=0[/math] and [math]P(\Omega)=1[/math]
(c) If [math]\{A_1,A_2,A_3,\cdots \}[/math] is a sequence of disjoint sets (i.e. [math]A_i \cap A_j = \emptyset[/math] whenever [math]i\neq j[/math]) that belong to [math]\mathcal{F}[/math], then [math]P(\cup_iA_i)= \sum^{\infty}_{i=1} P(A_i)[/math].



  • (Wikipedia, 2015) ⇒ Retrieved:2015-6-4.
    • In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. [1] The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space.

      Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. the value assigned to "1 or 2" in a throw of a die should be the sum of the values assigned to "1" and "2".

      Probability measures have applications in diverse fields, from physics to finance and biology.

  1. An introduction to measure-theoretic probability by George G. Roussas 2004 ISBN 0-12-599022-7 page 47




(a) [math]\Omega[/math] is the sample space, the set of possible outcomes of the experiment.
(b) [math]\mathcal{F}[/math] is a σ-field, a collection of subsets of [math]\Omega[/math].
(c) [math]\mathbb{P}[/math] is a probability measure, a function that assigns a nonnegative probability to every set in the σ-field F.
(...) Let [math](\Omega, \mathcal{F})[/math] be a measurable space. A measure is a function [math]\mu \; :\; \mathcal{F} \rightarrow [0, +\infty][/math], which assigns a nonnegative extended real number [math]\mu(A)[/math] to every set [math]A[/math] in [math]\mathcal{F}[/math], and which satisfies the following two conditions:
(a) [math]\mu(\emptyset)=0[/math];
(b) (Countable additivity) If [math]\{A_i\}[/math] is a sequence of disjoint sets that belong to [math]\mathcal{F}[/math], then [math]\mu(\cup_iA_i) = \sum^{\infty}_{i=1} \mu(A_i)[/math].
A probability measure is a measure [math]\mathbb{P}[/math] with the additional property [math]\mathbb{P}(\Omega)= 1[/math]. In that case, the triple \[math](\Omega, \mathcal{F}, \mathbb{P})[/math] is called a probability space.




  • (Kolmogorov, 1933) ⇒ Andrei Kolmogorov. (1933). “Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability.). American Mathematical Society. ISBN:0828400237
    • QUOTE: … Every distributions function [math]F_{\mu_1 \mu_2 … \mu_n}[/math], satisfying the general conditions of Chap. II, Sec 3, III and also conditions (2) and (3). Every distribution function [math]F_{\mu_1 \mu_2 … \mu_n}[/math] defines uniquely a corresponding probability function [math]\text{P}_{\mu_1 \mu_2 … \mu_n}[/math] for all Borel sets of [math]R^n[/math].