# Defining the integral of a function using the product measure

Imagine that we're trying to define the expression $$\int_U f(x)dx$$ in a rigorous way. Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a measurable subset of $X$. That most typical approach to making this integral rigorous is the method of Lebesgue, whereby we partition the range of $f$ into increasingly small horizontal strips. This seems very elaborate to me - why not just define the integral in the obvious way as the "(product) measure of the set of all points under the curve"? (if its defined; our integrable functions would then be precisely those for which the product measure is indeed defined). We can make this idea precise by writing

$$\int_U f(x)dx := (\mu \times \lambda)(\lbrace (x,y) : x \in U \wedge 0 \leq y \leq f(x)\rbrace)$$

where $\mu$ is the measure on $X$ and $\lambda$ is the standard measure on $\mathbb{R}$.

My question is, why isn't this the "standard" definition of the integral?

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You don't even need the construction of the product measure, even for integrals on a general measure space $(X,\mu)$: you may define the integral of a measurable $f:X\to[0,\infty]$ as the (Riemann) integral of its distribution function (a decreasing function): $$\int_X f(x) d\mu(x)=\int_0^\infty\mu\{f>t\}dt .$$ But the same remark in Jochen's answer holds, even for the simple case of the Lebesgue measure on $[0,1]$: doing something out of this definition turns out to be quite hard. –  Pietro Majer Mar 24 at 17:21

Linearity of this integral is very mysterious. Moreover, the definition of the product measure using integration, i.e. $\mu \otimes \lambda (M) =\int \int I_M(x,y) d\mu(x) d\lambda(y)$, is very easy (up to a technical problem concerning measurability) and can be understood without knowing Caratheodory's construction of measures.

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I'm not sure I buy Jochen's comment that product measure can be so easily defined using integration --- it seems like you're going to have to do some work to show that his double integral is well-defined for every set $M$ in the $\sigma$-algebra generated by the measurable rectangles.
The real problem may be that you actually "need" integration theory to define product measures via the standard Caratheodory construction, when you show that $(\mu\times\nu)(A \times B) = \mu(A)\nu(B)$ defines a premeasure on the algebra generated by the measurable rectangles. That is, if $A \times B$ can be expressed as a disjoint union $\bigcup A_i \times B_i$, we need $\mu(A)\nu(B) = \sum \mu(A_i)\nu(B_i)$. And as far as I can see you pretty much have to use the monotone convergence theorem to prove that.
Yes, it is not obvious that the iterated integral is well defined. But this is a technical problem which (for $\sigma$-finite measures) can be solved e.g. using Dynkin systems and, as you said, the monotone convergence theorem. However, imagine an average student in a course on probability theory: He could disregard all existence and measurability problems and nevertheless understand the proofs of almost all important results including the construction of product measures. –  Jochen Wengenroth Aug 31 '12 at 6:34