1. Lebesgue Measure and Integration
2. Properties of Outer Measure and Measurable Sets
3. Lemmas 7.4.2 and 7.4.4
4. Lemmas 7.4.6 - 7.4.11
5. Measurable Function, Regularity
6. Product and Slices
7. Pugh Lebesgue Integral 1
8. Pugh Lebesgue Integral 2
9. Tao Lebesgue Integral 1
10. Tao Lebesgue Integral 2

HW1 (Updated 1/26)
HW2 (Updated 2/3)
HW3 (Updated 2/11)
HW4 (Updated 2/17)
HW5 (Updated 2/25)
HW6 (Updated 3/2)
HW7 (Updated 3/11)
HW8 (Updated 3/18)
HW9 (Updated 3/25)
HW10 (Updated 4/6)
HW11 (Updated 4/16)
HW12 (Updated 4/23)

The Lebesgue integral is defined in terms of the undergraph. For a function $f: \mathbb{R}\rightarrow [0,\infty)$, the undergraph of $f$ is defined as $$u(f)=\{(x,y):0\leq y <f(x)\}$$ If $u(f)$ is measurable, we say that $f$ is measurable, and define the Lebesgue integral of $f$ as $$\int f = m(u(f))$$ If $\int f < \infty$, we say that $f$ is Lebesgue integrable. From this definition, we notice that the Lebesgue integral heavily depends on the notion of measure and measurability; this mode of integration has many benefits compared to the Riemann integral.

The Riemann integral is defined in terms of upper and lower integrals. For a function $f: [a,b] \rightarrow \mathbb{R}$, the upper and lower sums are defined as $$U(f,P)=\sum_{k=1}^{n}M_k (f)(x_k - x_{k-1})$$ $$L(f,P)=\sum_{k=1}^{n}m_k (f)(x_k - x_{k-1})$$ where $P$ is a partition of hte set $[a,b]$ and $$M_k (f) = sup \{f(x):x \in [x_{k-1},x_k] \}$$ $$m_k (f) = inf \{f(x):x \in [x_{k-1},x_k] \}$$ Then, the upper and lower integrals of $f$ are defined as $$U(f)=inf\{U(f,P): \text{P is a partition of [a,b]}\}$$ $$L(f)=sup\{L(f,P): \text{P is a partition of [a,b]}\}$$ Lastly, the Riemann integral over $[a,b]$ is defined as $$\int_{a}^{b}f(x)dx = L(f)=U(f)$$ From this definition, we notice that the notion of Riemann integral heavily depends on the interval on which it is defined. This could generate concerns over the flexibility of the Riemann integral that Lebesgue integral can handle more elegantly. For example, when both types of integrals are generalized to higher dimensions, the notion of an interval in Riemann integral is more difficult to define since we can have “irregular shapes” such as circles. On the other hand, Lebesgue integral can compute the measure of the undergraph defined on these “irregular shapes” more easily using box covers. I do think, however, measure is a rather abstract concept, its computation is a bit more difficult to understand compared to the “area under the curve” definition of the Riemann integral.

A paper that constructs a set which includes points at which the density of the set can take on any values in $[0,1]$