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?
Best Answer
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.