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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:
Riemann integrals are only defined for bounded functions over bounded intervals. Now, with Lebesgue integrals, we can integrate any measurable function.
Summary of Key Steps of Results in Lebesgue Measure Theory
