The Unapologetic Mathematician

Mathematics for the interested outsider

Double and Iterated Integrals

Let (X,\mathcal{S},\mu) and (Y,\mathcal{T},\nu) be two \sigma-finite measure spaces, and let \lambda=\mu\times\nu be the product measure on X\times Y.

If h is a function on X\times Y so that its integral is defined — either h is integrable, or its integral diverges definitely — then we write it as any of

\displaystyle\int h\,d\lambda=\int h\,d(\mu\times\nu)=\int h(x,y)\,d\lambda(x,y)=\int h(x,y)\,d(\mu\times\nu)(x,y)

and call it the “double integral” of h over X\times Y. We can also consider the sections h_x(y)=h(x,y) and h^y(x)=h(x,y). For any given x\in X, we set

\displaystyle\int h_x(y)\,d\nu(y)=f(x)

if the integral exists. If the resulting function f is integrable, then we write

\displaystyle\int f\,d\mu=\iint h(x,y)\,d\nu(y)\,d\mu(x)=\iint h\,d\nu\,d\mu=\int\,d\mu(x)\int h(x,y)\,d\nu(y)

The latter notation, with the measure \mu before the integrand is less common, but it can be seen in older texts. I’ll usually stick to the other order.

Similarly, we can define the function g(y) as the integral of the Y-section h^y if it exists. If g is integrable, we write

\displaystyle\int g\,d\nu=\iint h(x,y)\,d\mu(x)\,d\nu(y)=\iint h\,d\mu\,d\nu=\int\,d\nu(y)\int h(x,y)\,d\mu(x)

where, again, the latter notation is deprecated. These integrals are called the “iterated integrals” of h. We can also define double and iterated integrals over a measurable subset E\subseteq X\times Y, as usual, and write

\displaystyle\begin{aligned}\int\limits_Eh&\,d\lambda\\\int\limits_Eh&\,d(\mu\times\nu)\\\iint\limits_Eh&\,d\mu\,d\nu\\\iint\limits_Eh&\,d\nu\,d\mu\end{aligned}

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July 27, 2010 - Posted by | Analysis, Measure Theory

2 Comments »

  1. [...] if is a non-negative measurable function on , then we have the following equalities between the double integral and the two iterated [...]

    Pingback by Fubini’s Theorem « The Unapologetic Mathematician | July 28, 2010 | Reply

  2. [...] can turn this into an iterated integral, which Fubini’s theorem tells us we can evaluate in any order we [...]

    Pingback by Stokes’ Theorem (proof part 1) « The Unapologetic Mathematician | August 18, 2011 | Reply


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