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