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sigma-algebras

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Sigma Algebras

When to Use

Use this skill when working on sigma-algebras problems in measure theory.

Decision Tree

  1. Verify sigma-algebra axioms

    • X in F (whole space is measurable)
    • A in F implies A^c in F (closed under complements)
    • A_n in F implies union(A_n) in F (closed under countable unions)
    • z3_solve.py prove "sigma_algebra_axioms"

  2. sigma-algebra generation

    • Start with generating collection C
    • sigma(C) = smallest sigma-algebra containing C
    • Use Dynkin's pi-lambda theorem for uniqueness

  3. Measurability verification

    • f is measurable if f^{-1}(B) in F for all Borel B
    • Sufficient: check for open sets or intervals
    • sympy_compute.py simplify "preimage(f, interval)"

  4. Product sigma-algebras

    • F1 x F2 = sigma{A x B : A in F1, B in F2}
    • Projections are measurable

Tool Commands

Z3_Sigma_Axioms

uv run python -m runtime.harness scripts/z3_solve.py prove "X_in_F and closed_under_complement and closed_under_countable_union"

Z3_Dynkin_Pi_Lambda

uv run python -m runtime.harness scripts/z3_solve.py prove "pi_system_subset_lambda implies sigma_equal"

Sympy_Preimage

uv run python -m runtime.harness scripts/sympy_compute.py simplify "f_inv(A_union_B) == f_inv(A) | f_inv(B)"

Key Techniques

From indexed textbooks:

  • [Statistical Inference (George Casella... (Z-Library)] PROBABILITY THEORY Definition 1. A collection of subsets of S is called a sigma algebra (or Borel field), denoted by B, if it satisfies the following three properties: a. B (the empty set is an element of B).
  • [Measure, Integration Real Analysis (... (Z-Library)] S T is the smallest s-algebra containing the measurable rectangles). S T The technique outlined above should be used when possible. However, in some situations there seems to be no reasonable way to verify that the collection of sets with the desired property is a s-algebra.
  • [Statistical Inference (George Casella... (Z-Library)] Thus, again by property (b), N2, A; € B. Associated with sample space § we can have many different sigma algebras. For example, the collection of the two sets {#, S} is a sigma algebra, usually called the trivial sigma algebra.
  • [Real Analysis (Halsey L. Royden, Patr... (Z-Library)] Proposition 13 Let F be a collection of subsets of a set X. Then the intersection A of all σ-algebras of subsets of X that contain F is a σ-algebra that contains F. Moreover, it is the smallest σ-algebra of subsets of X that contains F, in the sense that any σ-algebra that contains F also contains A.
  • [Real Analysis (Halsey L. Royden, Patr... (Z-Library)] Let M be the collection of subsets of X that are either countable or have a countable complement in X. For E ∈ M, dene µ(E) = 0 if E is countable and µ(E) = 1, if E has a countable complement. Is this measure space complete?

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.

Source

git clone https://github.com/parcadei/Continuous-Claude-v3/blob/main/.claude/skills/math/measure-theory/sigma-algebras/SKILL.mdView on GitHub

Overview

This skill guides you through core sigma-algebra tasks in measure theory: verify axioms, generate the smallest sigma-algebra from a collection, and verify measurability and product sigma-algebras. It also highlights using Dynkin's pi-lambda theorem for uniqueness and automating proofs with Z3 and SymPy tools.

How This Skill Works

Follow a decision-tree workflow: 1) verify the sigma-algebra axioms for a candidate F; 2) generate sigma(C) as the smallest sigma-algebra containing a generating collection C; 3) verify measurability by checking f^{-1}(B) for Borel sets (sufficient to test open sets or intervals); 4) treat product spaces via F1 x F2 = sigma{A x B : A in F1, B in F2} and confirm projections are measurable. Tool commands automate proofs: Z3_Sigma_Axioms, Z3_Dynkin_Pi_Lambda, and Sympy_Preimage assist with respective steps.

When to Use It

  • You need to prove a candidate collection F is a sigma-algebra on X.
  • You must construct the smallest sigma-algebra containing a given family C.
  • You want to verify a function is measurable by checking preimages of Borel sets.
  • You are working with product spaces and need the product sigma-algebra and measurability of projections.
  • You need to establish uniqueness of a sigma-algebra containing a generating collection using pi-lambda arguments.

Quick Start

  1. Step 1: Decide if you’re proving a sigma-algebra or generating one from a collection C.
  2. Step 2: Use sigma(C) = smallest sigma-algebra containing C and, if needed, apply Dynkin's pi-lambda for uniqueness.
  3. Step 3: For measurability or products, verify preimages of open/interval sets and confirm projections are measurable.

Best Practices

  • Start by verifying sigma-algebra axioms: X in F, closed under complements, and closed under countable unions.
  • Identify a generating collection C and compute sigma(C) as the smallest sigma-algebra containing C.
  • When possible, apply Dynkin's pi-lambda theorem to simplify uniqueness arguments.
  • Leverage tool commands to automate proofs: Z3_Sigma_Axioms, Z3_Dynkin_Pi_Lambda, and Sympy_Preimage.
  • For measurability checks, test preimages of open sets or intervals; use common cases like singletons and standard Borel sets.

Example Use Cases

  • Prove the axioms for the trivial or full sigma-algebra on a finite space.
  • Construct the Borel sigma-algebra on R from open intervals.
  • Generate a sigma-algebra from a given collection of sets and verify it is minimal.
  • Check measurability of a function f: R -> R by verifying f^{-1}(I) is measurable for intervals I.
  • Build the product sigma-algebra on R×R and show that projections pi1, pi2 are measurable.

Frequently Asked Questions

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