A student claims: 'The product measure is fully defined by the formula (μ⊗ν)(A×B) = μ(A)·ν(B), so there's nothing more to construct once we specify what it does on rectangles.' What is wrong with this claim?
ANothing is wrong — the rectangle formula completely defines (μ⊗ν) on all of ℱ⊗𝒢
BMeasurable rectangles don't cover all sets in ℱ⊗𝒢, so the Carathéodory extension theorem is needed to extend the measure to non-rectangular sets
CThe product σ-algebra ℱ⊗𝒢 contains only sets of the form A×B, so rectangles do cover everything
DThe formula μ(A)·ν(B) is undefined unless both μ(A) and ν(B) are finite
Measurable rectangles A×B are merely the generators of ℱ⊗𝒢 — the full σ-algebra contains complements, countable unions, and intersections of rectangles, most of which are not themselves rectangles (think of diagonal strips, fractal regions, or complements of rectangles in X×Y). The Carathéodory extension theorem guarantees a unique extension of the rectangle assignment to all of ℱ⊗𝒢, and that uniqueness is essential: it tells us there is exactly one product measure consistent with the rectangle formula.
Question 2 Multiple Choice
Which statement correctly describes the product σ-algebra ℱ⊗𝒢?
AIt consists exactly of sets of the form A×B where A∈ℱ and B∈𝒢
BIt is generated by measurable rectangles A×B, but contains many sets that are not themselves rectangles
CIt contains all subsets of X×Y whenever ℱ and 𝒢 are Borel σ-algebras
DIt is defined as the union of ℱ and 𝒢 applied to the coordinates of X×Y separately
A σ-algebra generated by a collection of sets is closed under complements and countable unions, so it contains far more than just the generating sets. The complement of A×B in X×Y is not generally a rectangle, and neither are countable unions of rectangles. Option A is the most common misconception — conflating the generators of ℱ⊗𝒢 with the σ-algebra itself. Option C is false: even the Borel σ-algebra on ℝ² does not contain all subsets of ℝ².
Question 3 True / False
The product measure (μ⊗ν) is the unique measure on ℱ⊗𝒢 satisfying (μ⊗ν)(A×B) = μ(A)·ν(B) for all measurable rectangles A×B (assuming σ-finiteness).
TTrue
FFalse
Answer: True
This is exactly what the Carathéodory extension theorem guarantees: uniqueness, not just existence. The σ-finiteness assumption on μ and ν is needed for uniqueness — without it, multiple extensions may exist. Uniqueness matters because it means when Fubini's theorem invokes the product measure, there is no ambiguity about which measure is meant.
Question 4 True / False
Most set in the product σ-algebra ℱ⊗𝒢 can be written as a finite union of measurable rectangles A×B.
TTrue
FFalse
Answer: False
The σ-algebra generated by rectangles is closed under countable (not just finite) unions and complements. Sets like {(x,y) : x+y < 1} in ℝ² require infinitely many rectangles to approximate, and their complements are not finite unions of rectangles either. This is precisely why the Carathéodory extension — which handles the full machinery of measure theory — is needed, not just a finite combinatorial argument.
Question 5 Short Answer
Why can't the product measure (μ⊗ν) be fully defined just by specifying its value on measurable rectangles, and what theorem completes the construction?
Think about your answer, then reveal below.
Model answer: Measurable rectangles only generate the product σ-algebra — they don't constitute it. There are sets in ℱ⊗𝒢 (like complements of rectangles or countable unions of rectangles) that are not themselves rectangles, so the rectangle formula alone leaves the measure undefined on most of ℱ⊗𝒢. The Carathéodory extension theorem fills the gap: it extends a pre-measure defined on a generating semiring (here, the measurable rectangles) to a full measure on the generated σ-algebra, and guarantees the extension is unique under σ-finiteness.
The key insight is that defining a measure on generators is not the same as defining it on the whole σ-algebra. The Carathéodory extension is the rigorous bridge. Its uniqueness is what makes product measures well-defined and allows Fubini's theorem to work without ambiguity about which measure is being iterated.