Measure, Integration & Real Analysis, 2021a

Chapter 5
Product Measures
Lebesgue measure on R generalizes the notion of the length of an interval. In this
chapter, we see how two-dimensional Lebesgue measure on R 2 generalizes the notion
of the area of a rectangle. More generally, we construct new measures that are the
products of two measures.
Once these new measures have been constructed, the question arises of how to
compute integrals with respect to these new measures. Beautiful theorems proved in
the first decade of the twentieth century allow us to compute integrals with respect to
product measures as iterated integrals involving the two measures that produced the
product. Furthermore, we will see that under reasonable conditions we can switch
the order of an iterated integral.
Main building of Scuola Normale Superiore di Pisa, the university in Pisa, Italy,
where Guido Fubini (1879–1943) received his PhD in 1900. In 1907 Fubini proved
that under reasonable conditions, an integral with respect to a product measure can
be computed as an iterated integral and that the order of integration can be switched.
Leonida Tonelli (1885–1943) also taught for many years in Pisa; he also proved a
crucial theorem about interchanging the order of integration in an iterated integral.
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Measure, Integration & Real Analysis, by Sheldon Axler
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