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Basic•Statistics
Probability
Understanding randomness and Bayesian vs Frequentist views.
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Research-Level Deep Dive & Equations
Classical probability theory often assumes that we can assign a probability to any arbitrary subset of the sample space . While this works for finite or countably infinite sample spaces (where the power set serves as the domain), it breaks down completely when is uncountable, such as the real line or the unit interval . Under the Axiom of Choice, trying to define a translation-invariant, countably additive probability measure on the entire power set leads to mathematical contradictions, such as the existence of non-measurable sets (e.g., the Vitali set) and paradoxical decompositions like the Banach-Tarski paradox.
To resolve this, we restrict the domain of our probability measure to a special collection of subsets called a -algebra (sigma-algebra). Let be a non-empty sample space. A collection of subsets of is called a -algebra on if it satisfies the following three foundational axioms:
1. **Contains the Empty Set**: (which, combined with closure under complements, implies ).
2. **Closure under Complementation**: If an event , then its complement .
3. **Closure under Countable Unions**: If a sequence of events , then their union .
Using De Morgan's laws, we can easily prove that -algebras are also closed under countable intersections:
On the real line , the standard -algebra is the **Borel -algebra**, denoted . It is defined as the smallest -algebra containing all open sets (or equivalently, all open intervals ) in . Subsets of that belong to are called Borel sets. Borel sets encompass virtually all subsets encountered in practical analysis, including all open, closed, semi-open intervals, and their countable unions and intersections. The pair is called a **measurable space**, and elements are called **measurable sets** or **events** .
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