Measure Theory 17 | Product measure and Cavalieri's principle [dark version]

Product measure turns two separate measure spaces into a single measure on their Cartesian product by enforcing a simple rule: measurable rectangles must have measure equal to “length times width.” Starting with measure spaces (X1, A1, μ1) and (X2, A2, μ2), the construction lives on X1 × X2. One chooses the product σ-algebra generated by measurable rectangles A1 × A2, and then defines the product measure μ so that for every measurable rectangle, μ(A1 × A2) = μ1(A1)·μ2(A2). This rectangle rule is the core requirement; it pins down how “volume” should behave in the coordinate system where sets look like blocks.

Defining μ on rectangles is only the beginning, because most interesting subsets of X1 × X2 are not rectangles. The transcript illustrates this with a triangle-shaped set M. To compute μ(M), the method uses sections: fix a point y in X2 and look at the “slice” of M along the X1-direction, denoted M_y = {x ∈ X1 : (x, y) ∈ M}. The measure of that slice is μ1(M_y). Then the overall measure of M is obtained by aggregating all slices across X2—conceptually, summing the red vertical slices. In measure-theoretic terms, this aggregation is expressed as an integral over X2 with respect to μ2: μ(M) is computed by integrating the slice-measure μ1(M_y) over y.

A symmetric statement holds when slicing in the other direction: for fixed x ∈ X1, consider the slice M^x = {y ∈ X2 : (x, y) ∈ M}, measure it using μ2, and integrate over X1 with respect to μ1. This “slice-and-integrate” viewpoint is exactly what Cavalieri’s principle looks like in the language of product measures: volume of a general set is built from the measures of its one-direction sections.

The construction also raises a subtlety about existence and uniqueness. Using an extension theorem (via defining μ on a semiring of rectangles and extending to the generated σ-algebra), the product measure can be built, but the extension need not be unique in full generality. Uniqueness becomes guaranteed under the condition that both μ1 and μ2 are σ-finite—an assumption familiar from standard Lebesgue measure on R^n. Under σ-finiteness, there is exactly one product measure satisfying the rectangle rule.

In short, the transcript connects two ideas: the product measure is defined by enforcing the rectangle formula, and Cavalieri’s principle is the practical way to compute the measure of non-rectangular sets by integrating the measures of their slices in one coordinate direction and summing them across the other.