Lecture Notes

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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 $\int f = m(U(f))$.