This is an old revision of the document!
Summary of Lebesgue Integral
We define a Lebesgue Integral by introducing the notion of an undergraph of a function $f:\mathbb{R}\mapsto [0, \infty]$. We define the undergraph $U(f)$ as $\{(x, y) \in \mathbb{R} \times [0, \infty]: 0 \leq y < f(x) \}$.
We say that $f$ is measurable if $U(f)$ is measurable, and if this is the case, then we can officially define the Lebesgue Integral as $\int f = m(U(f))$.
To compare and contrast Lebesgue integration with Riemann integration, we can consider a number of different aspects:
One significant difference is that the Riemann integral can only integrate over a finite number of discontinuities, whereas the Lebesgue integral can handle an infinite number of discontinuities. Riemann integrals are only defined for bounded functions over bounded intervals. Now, with Lebesgue integrals, we can integrate any measurable function.
