Probability Space

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A Probability Space is a Set Measure Space in which the Set Measure Function (the Probability Measure) over the entire Measurable Space (the Sample Space) equals One (μ(S)=1 or P(Ω)=1).


  • (Wikipedia, 2009) ⇒
    • A probability space, in probability theory, is the conventional mathematical model of randomness. This mathematical object, sometimes called also probability triple, formalizes three interrelated ideas by three mathematical notions. First, a sample point (called also elementary event), --- something to be chosen at random (outcome of experiment, state of nature, possibility etc.) Second, an event, --- something that will occur or not, depending on the chosen elementary event. Third, the probability of this event. The definition (see below) was introduced by Kolmogorov in the 1930s. For an algebraic alternative to Kolmogorov's approach, see algebra of random variables. Alternative models of randomness (finitely additive probability, non-additive probability) are sometimes advocated in connection to various probability interpretations.
    • A probability space is a measure space such that the measure of the whole space is equal to 1.
    • In other words: a probability space is a triple (Ω, F, P) consisting of a set Ω (called the sample space), a σ-algebra (also called σ-field) F of subsets of Ω (these subsets are called events), and a measure P on (Ω, F) such that P(Ω)=1 (called the probability measure).
    • Discrete Example 1: If the space concerns one flip of a fair coin, then the outcomes are heads and tails: Ω = {H,T}. The σ-algebra F = 2^Ω contains 2^2 = 4 events, namely, {H} : heads, {T} : tails, {} : neither heads nor tails, and {H,T} : heads or tails. So, F = { {}, {H}, {T}, {H,T}}. There is a fifty percent chance of tossing either heads or tail: p(H) = p(T) = 0.5; thus P({H}) = P({T}) = 0.5. The chance of tossing neither is zero: P({})=0, and the chance of tossing one or the other is one: P({H,T})=1.
    • Discrete Example 2: The fair coin is tossed 3 times. There are 8 possibilities: Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} . The complete information is described by the σ-algebra F = 2^Ω of 2^8 = 256 events (just one of them: {HTT, THT}).


  • (Qian) => Gang Qian. (2008). Basic Probability Theory." Lecture Notes: AME 598 Sensor Fusion, Arizona State University, Fall 2008.
    • Probability Space {S,E,P}
      • S: Sample space, the space of the outcomes from a random experiment {o}
      • E: Event space, collection of subsets of the sample space, {A}
      • P: Probability measure of a event P(A), ranges [0,1], encoding how likely an event will happen.