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ISEGORIABenjamin Haire
日本語

Visual mathematics · Interactive lab + 13-page PDF

Lebesgue integration, from measure to calculation.

I teach the Lebesgue integral by contrast with the Riemann integral, and I lead with diagrams: measurable sets, simple functions, signed integrals, convergence theorems, Tonelli and Fubini, with seven fully worked examples and a final practice set.

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The construction

The order I build it in
01Measurable sets
02Indicators
03Simple functions
04Nonnegative limits
05Signed functions

Inside the guide

Theory + method + examples
  1. Partition the range. Riemann versus Lebesgue.
  2. Measure spaces. Sigma-algebras, measure, null sets.
  3. Measurable functions. Preimages and simple functions.
  4. The definition. Approximation from below.
  5. Calculation workflow. Six practical routes.
  6. Foundational examples. Indicators and Dirichlet's function.
  7. Singularities and tails. Power tests and layer cake.
  8. Limit theorems. MCT, Fatou, and DCT.
  9. Worked limit exchanges. Domination and series.
  10. Product measure. Tonelli and Fubini.
  11. The L1 viewpoint. Norm and absolute continuity.
  12. Practice. Exercises, answers, and references.