Discrete Case
Discrete probability theory needs only at most countable sample spaces Ω. Probabilities can be ascribed to points of Ω by the probability mass function p: Ω→ such that ∑ω∈Ω p(ω) = 1. All subsets of Ω can be treated as events (thus, = 2Ω is the power set). The probability measure takes the simple form
The greatest σ-algebra = 2Ω describes the complete information. In general, a σ-algebra ⊆ 2Ω corresponds to a finite or countable partition Ω = B1 ⊔ B2 ⊔ ..., the general form of an event A ∈ being A = Bk1 ⊔ Bk2 ⊔ ... (here ⊔ means the disjoint union.) See also the examples.
The case p(ω) = 0 is permitted by the definition, but rarely used, since such ω can safely be excluded from the sample space.
Read more about this topic: Probability Space
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