Sigma-algebras and measure spaces

Mathematical statement

On a set X, a sigma-algebra contains the empty set and is closed under complements and countable unions. A measure assigns nonnegative values to measurable sets, is zero on the empty set and is countably additive on pairwise disjoint countable families. This slice’s complete finite carrier permits reduction to finitely many distinct sets.

Representative formula (exact TeX)

M(S)

Assumptions

• Exercises use a finite carrier, a complete canonical set family and exact nonnegative rational measure values.
• Finite reduction establishes only the specified finite measure space, not arbitrary infinite sigma-algebras or extended-real behavior.

Symbol scope

• M — finite-measure-space record
• S — measurable set family

Intuition, definition, and misconceptions

On a finite carrier, a canonical family can be fully checked for complement and binary-union closure. Finitely many distinct subsets then give the countable-union reduction.

Audit zero empty mass and finite additivity for every disjoint pair. A disjoint family on a finite carrier has only finitely many nonempty members.

A measure is a probability measure only when its total mass is 1.

Do not declare an arbitrary set family a sigma-algebra merely because it contains the empty and whole sets, or unconditionally extend finite additivity to infinite carriers.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Set — The carrier and measurable subsets are sets.
• Set operations — Complement and union closure use set operations.
• Function mapping — A measure maps measurable sets to nonnegative values.
• Sequence — Sequences index countable families.
• Quantifiers and explicit domains — Countable closure and additivity include universal obligations.
• Real number — Measure values in this finite slice are nonnegative reals.

Verification boundary

Countable semantics are closed by finite reduction only for the specified finite carrier.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add measurable functions, the Lebesgue integral and convergence theorems.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.