Measure Theory and HBD

I've recently learned measure theory from Tao's book and these really good course notes. Here is an example of the content: ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ $$ \textbf{Main Theorem}\qquad \text{Existence of Lebesgue Measure} $$ $$ \begin{aligned} &\text{There exists a collection } \mathcal{M} \text{ of subsets of } \mathbb{R} \text{ (the measurable sets) and a function} \\ &\qquad m:\mathcal{M}\to [0,\infty] \text{ satisfying the following conditions:} \end{aligned} $$ Every interval $I\subseteq \mathbb{R}$ is measurable, with $$ m(I)=\ell(I). $$ If $E\subseteq \mathbb{R}$ is a measurable set, then the complement $$ E^{c}=\mathbb{R}-E $$ is also measurable. For each sequence $\{En\}$ of measurable sets in $\mathbb{R}$, the union $$ \bigcup{n\in\mathbb{N}} En $$ is also measurable. Moreover, if the sets $\{En\}$ are pairwise disjoint, then $$ m\!\left(\bigcup{n\in\mathbb{N}} En\right)=\sum{n\in\mathbb{N}} m(En). $$ ------------------------------------------------------------------------------------------…