sigma-algebras

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