Product Measure and Fubini's Theorem
Two measure spaces combine into one, and an integral over the product can be done one variable at a time. The hypotheses of the theorem say exactly when the order of integration is free.
What a learner can do afterwards
- State the difference between the Tonelli and Fubini hypotheses
- Exchange the order of a double integral and justify the exchange
- Give an example where the two iterated integrals differ, and name the hypothesis that fails
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You combine two measure spaces into one product space, then integrate over both variables at once. Tonelli says you may split the double integral into iterated integrals when the function is nonnegative and measurable. Fubini says the same for a sign changing function, but only when the integral of its absolute value is finite.
Take f(x, y) = x times y on the rectangle with x from 0 to 2 and y from 0 to 3. Integrate in y first to get 4.5 times x, then in x to get 9, and the other order gives 9 too. The exchange is licensed because a continuous function on a closed rectangle is bounded, so its absolute integral is finite and Fubini applies. A constant function is even easier: its integral is the constant times the area.
Some sign changing functions have two iterated integrals that both exist yet give different answers. In each case the integral of the absolute value is infinite, so Fubini does not apply, and Tonelli cannot help because the function takes negative values. Treat any such mismatch as the signal that absolute integrability failed.
Before you exchange the order, run this check. If the function is nonnegative, Tonelli covers you. Otherwise compute the integral of the absolute value and confirm it is finite, and then Fubini covers you. Defined iterated integrals alone never license the exchange.
Tonelli needs nonnegativity, Fubini needs a finite absolute integral, and either one lets you integrate one variable at a time.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.