integration-theory

# Integration Theory ## When to Use Use this skill when working on integration-theory problems in measure theory. ## Decision Tree 1. **Simple function integration** - For s = sum(a_i * chi_{E_i}): integral s dmu = sum(a_i * mu(E_i)) - `sympy_compute.py simplify "simple_integral"` 2. **Monotone Convergence Theorem (MCT)** - If 0 <= f_n <= f_{n+1} and f_n -> f: - lim integral(f_n) = integral(lim f_n) - Use for increasing sequences 3. **Dominated Convergence Theorem (DCT)** - If |f_n| <= g (integrable) and f_n -> f pointwise: - lim integral(f_n) = integral(f) - `z3_solve.py prove "dominated_convergence"` 4. **Fatou's Lemma** - integral(liminf f_n) <= liminf(integral f_n) - Use as lower bound when MCT/DCT don't apply 5. **Fubini-Tonelli** - For product measures: switch order of integration - Tonelli: non-negative functions (always valid) - Fubini: integrable functions ## Tool Commands ### Sympy_Simple_Integral ```bash uv run python -m runtime.harness scripts/sympy_compute.py integrate "sum(a_i * chi_E_i)" --var mu ``` ### Z3_Mct ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "f_n_increasing implies lim_integral_equals_integral_lim" ``` ### Z3_Dct ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "abs(f_n) <= g and g_integrable implies limit_exchange" ``` ### Sympy_Fatou ```bash uv run python -m runtime.harness scripts/sympy_compute.py limit "liminf(integral_f_n)" --comparison "integral_lim...