Probability space (Ω, ℱ, P): sample space Ω, σ-algebra ℱ of events, probability measure P. Kolmogorov's three axioms (1933) are the standard foundation. Arrows mean "depends on".
Source: Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (1933); Billingsley, Probability and Measure
Dependency Flowchart
graph TD
D1["Def: Sample space Ω\nset of outcomes"]
D2["Def: σ-algebra ℱ\nclosed under complements,\ncountable unions"]
D3["Def: Event\nA ∈ ℱ"]
A1["A1 Non-negativity\nP(A) ≥ 0"]
A2["A2 Unit measure\nP(Ω) = 1"]
A3["A3 σ-additivity\nP(∪Aⱼ)=ΣP(Aⱼ)\ndisjoint"]
T1["Thm: P(∅) = 0"]
T2["Thm: P(Aᶜ) = 1 − P(A)"]
T3["Thm: P(A) ≤ 1"]
T4["Thm: Finite additivity"]
T5["Thm: Monotonicity\nA⊆B ⇒ P(A)≤P(B)"]
D4["Def: Random variable\nX: Ω → ℝ measurable"]
D5["Def: Independence\nP(A∩B)=P(A)P(B)"]
D1 --> D2
D2 --> D3
D1 --> A2
D2 --> A1
D2 --> A3
A1 --> T1
A2 --> T1
A3 --> T2
A1 --> T3
A2 --> T3
A3 --> T4
T2 --> T5
T4 --> T5
A1 --> D4
A2 --> D4
A3 --> D4
D3 --> D5
A3 --> D5
classDef axiom fill:#ff6b6b,color:#fff
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class A1,A2,A3 axiom
class D1,D2,D3,D4,D5 definition
class T1,T2,T3,T4,T5 theorem
Process Statistics
- Nodes: 18
- Edges: 22
- Axioms: 3
- Definitions: 5
- Theorems: 5
Frontier: Extensions to finitely additive probability, non-Kolmogorov frameworks (e.g. quantum), imprecise probability. math.PR