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integration-theory

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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

uv run python -m runtime.harness scripts/sympy_compute.py integrate "sum(a_i * chi_E_i)" --var mu

Z3_Mct

uv run python -m runtime.harness scripts/z3_solve.py prove "f_n_increasing implies lim_integral_equals_integral_lim"

Z3_Dct

uv run python -m runtime.harness scripts/z3_solve.py prove "abs(f_n) <= g and g_integrable implies limit_exchange"

Sympy_Fatou

uv run python -m runtime.harness scripts/sympy_compute.py limit "liminf(integral_f_n)" --comparison "integral_liminf_f_n"

Key Techniques

From indexed textbooks:

  • [Measure, Integration Real Analysis (... (Z-Library)] If you go at a leisurely pace, then covering Chapters 1–5 in the rst semester may be a good goal. If you go a bit faster, then covering Chapters 1–6 in the rst semester may be more appropriate. For a second-semester course, covering some subset of Chapters 6 through 12 should produce a good course.
  • [Measure, Integration Real Analysis (... (Z-Library)] Suppose B is a Borel set and f : B R is a Lebesgue measurable function. B : g(x) = f (x) gj Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4. International License (http://creativecommons.
  • [Measure, Integration Real Analysis (... (Z-Library)] Statue in Milan of Maria Gaetana Agnesi, who in 1748 published one of the rst calculus textbooks. A translation of her book into English was published in 1801. In this chapter, we develop a method of integration more powerful than methods contemplated by the pioneers of calculus.
  • [Measure, Integration Real Analysis (... (Z-Library)] Preface for Instructors Chapter 3: Integration with respect to a measure is dened in this chapter in a natural fashion rst for nonnegative measurable functions, and then for real-valued measurable functions. The Monotone Convergence Theorem and the Dominated Convergence Theorem are the big results in this chapter that allow us to interchange integrals and limits under appropriate conditions. Preface for Instructors Chapter 8: This chapter focuses on Hilbert spaces, which play a central role in modern mathematics.
  • [Measure, Integration Real Analysis (... (Z-Library)] Chapter 6: After a quick review of metric spaces and vector spaces, this chapter denes normed vector spaces. The big result here is the Hahn–Banach Theorem about extending bounded linear functionals from a subspace to the whole space. Then this chapter introduces Banach spaces.

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/integration-theory/SKILL.mdView on GitHub

Overview

Provides problem-solving strategies for core integration theory results in measure theory, including simple function integration, Monotone Convergence Theorem (MCT), Dominated Convergence Theorem (DCT), Fatou's Lemma, and Fubini-Tonelli. It helps identify the right theorem, apply standard calculations, and verify when limits can be interchanged with integrals.

How This Skill Works

It guides you through a decision tree: decompose a target function into a simple form or recognize a converging sequence, then apply the appropriate theorem under its hypotheses. It also points to command templates (Sympy, Z3, etc.) to verify equalities or prove convergence.

When to Use It

  • When solving simple function integrals of the form s = sum a_i chi_{E_i}.
  • When you have an increasing sequence f_n with f_n -> f and 0 <= f_n.
  • When |f_n| <= g for some integrable g and f_n -> f pointwise.
  • When MCT/DCT don’t apply and you need a Fatou-type lower bound.
  • When dealing with product measures and you need to switch the order of integration (Tonelli/Fubini).

Quick Start

  1. Step 1: Express your target as s = sum a_i chi_{E_i}.
  2. Step 2: Determine which theorem applies (MCT, DCT, Fatou, or Fubini-Tonelli).
  3. Step 3: Compute the integral or justify interchange using the chosen theorem.

Best Practices

  • Write the target function as a simple function s = sum a_i chi_{E_i} to get the integral as sum a_i mu(E_i).
  • Check monotonicity and convergence hypotheses before invoking MCT or DCT.
  • Use Fatou’s Lemma as a fallback lower bound when convergence theorems fail.
  • For product measures, verify non-negativity or integrability before applying Tonelli or Fubini.
  • Use the provided tool commands to validate steps and results (Sympy_Simple_Integral, Z3_Mct, etc.).

Example Use Cases

  • Compute the integral of a simple function s = sum a_i chi_{E_i} by summing a_i mu(E_i).
  • Apply MCT for an increasing sequence f_n to justify exchange of limit and integral.
  • Use DCT with a dominating function g to pass the limit under the integral sign.
  • Apply Fatou’s Lemma to obtain a lower bound when MCT/DCT aren’t applicable.
  • Use Fubini-Tonelli to swap integrals in a product-measure setting.

Frequently Asked Questions

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