Given a measurable space $(X,\mathcal{T},\mu)$ and a measurable function $f:X\to\mathbb{R}_+$, the Lebesgue integral of $f$ is usually defined as the least upper bound of the integrals of finite linear combinations of indicator functions of measurable sets below $f$ (that's a lot of "of", sorry).
When $X$ is $\sigma$-finite, we have the (outer) product measure $\mu\times\lambda$ with the Lebesgue measure, and there seems to be a more direct definition of the integral of $f$,
$$ (\mu\times\lambda)(\{(x,y)\in X\times\mathbb{R}_+ \;/\; y \leq f(x) \} )$$
which is simply the volume below the graph of $f$.
Are these two definitions equal ?
Best Answer
Yes. All you have to do is evaluate the volume using Fubini's Theorem, (integrating w.r.t. $y$ first).