Chapter 7 extends the one-dimensional construction to two dimensions—just far enough to support Fubini’s theorem and practical computations.

7.1 Measure of a rectangle

The entry point is the rectangle:

  • Define the measure of a rectangle using the product of one-dimensional measures (generated by α in each coordinate).

This is the seed of product measure: the notion that “size in 2D” should factor when sets factor.

7.2 Simple sets and simple functions in two dimensions

As in 1D, you start with a tractable class:

  • simple sets built from rectangles,
  • simple functions built from indicator functions of those sets.

Then you define the integral on these objects and extend by approximation.

7.3 The Lebesgue–Stieltjes double integral

The double integral is defined in the same spirit:

  • integrate nonnegative simple functions,
  • then take limits via monotone approximation,
  • then extend to signed functions where appropriate.

7.4 Repeated integrals and Fubini’s theorem

This is the payoff: conditions under which

  • ∬ f(x, y) dμ(x, y)

can be computed as an iterated integral:

  • ∫ ( ∫ f(x, y) dμ₁(x) ) dμ₂(y)

and the order of integration can be swapped.

The operational point:

  • Fubini/Tonelli style results are what let you actually compute multi-variable integrals and justify exchanging summation/integration.

Mental model

The 2D extension is not about general measure theory on ℝ²; it’s about preserving the two essential benefits:

  • approximation by simple objects,
  • convergence theorems strong enough to justify computation by iterated integration.

Practice checklist

  • Work through examples where one coordinate has atomic mass (jumps in α) and see the integral become a mix of sums and integrals.
  • Practice conditions for swapping integration order: nonnegativity vs domination.