lebesgue-measure

# Lebesgue Measure ## When to Use Use this skill when working on lebesgue-measure problems in measure theory. ## Decision Tree 1. **Outer measure construction** - m*(A) = inf{sum |I_n| : A subset union(I_n)} - `sympy_compute.py sum "length(I_n)" --var n` 2. **Caratheodory criterion** - E is measurable if: m*(A) = m*(A & E) + m*(A & E^c) for all A - `z3_solve.py prove "caratheodory_criterion"` 3. **Lebesgue measure properties** - Translation invariant: m(E + x) = m(E) - sigma-additive on measurable sets - m([a,b]) = b - a 4. **Regularity theorems** - Inner regularity: m(E) = sup{m(K) : K compact, K subset E} - Outer regularity: m(E) = inf{m(U) : U open, E subset U} ## Tool Commands ### Sympy_Outer_Measure ```bash uv run python -m runtime.harness scripts/sympy_compute.py sum "length(I_n)" --var n --from 1 --to oo ``` ### Z3_Caratheodory ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "mu(A) == mu(A & E) + mu(A & E_complement)" ``` ### Sympy_Borel_Sets ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "open_set_countable_union" ``` ## Key Techniques *From indexed textbooks:* - [Measure, Integration Real Analysis (... (Z-Library)] Lebesgue measure on the Lebesgue measurable sets does have one small advantage over Lebesgue measure on the Borel sets: every subset of a set with (outer) measure 0 is Lebesgue measurable but is not necessarily a Borel set. However, any natural process that produces ...