\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
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\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)
Math and science::Analysis::Tao, measure::02. Lebesgue measure

The 3 basic properties of Lebesgue outer measure 

Reminder that the Lebesgue outer measure is denoted as \( m^{*} \).

The 3 basic propositions of Lebesgue outer measure

Empty set
[...]
Monotonicity
If \( E  \subseteq F \subset \mathbb{R}^d \), then [...].
Countable subadditivity
If \( E_1, E_2, ... \subset \mathbb{R}^d \) is a countable sequence of sets, then [...]

These three ideas are very fundamental. The complex apprearance of the expressions obscures the simplicity of the ideas.