Product measures
Objective. Use Tonelli and Fubini theorems.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Product measures combine two measure spaces (X,𝒜,μ) and (Y,ℬ,ν) into (X×Y,𝒜⊗ℬ,μ×ν), where the product σ-algebra is generated by rectangles A×B and the measure satisfies (μ×ν)(A×B)=μ(A)ν(B). Tonelli's theorem allows computing double integrals of nonnegative measurable functions as iterated integrals in either order without extra hypotheses, while Fubini's theorem extends this to functions integrable in absolute value. For instance, ∫∫_{[0,1]²} xy dx dy can be computed as ∫₀¹(∫₀¹xy dx)dy = ∫₀¹(y/2)dy = 1/4, and switching the order gives the same value.
Worked example
Using a product measure, evaluate ∬_{[0,2]×[0,3]} xy d(x,y).
- 1Apply Tonelli since the integrand is nonnegative on this rectangle
- 2∬xy = (∫₀²x dx)(∫₀³y dy)
- 3∫₀²x dx = 2, ∫₀³y dy = 4.5
- 4Product = 2 × 4.5 = 9
Answer: 9
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Start practisingCommon mistake. Students swap integration order freely without checking integrability of |f|; for f(x,y)=(x²−y²)/(x²+y²)² on (0,1)², the iterated integrals exist but differ, since f is not absolutely integrable, so Fubini does not apply. Verify ∫∫|f|<∞ first.
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