Pre-measure/Properties in common with measure

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$

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• Finitely additive: if $A\cap B=\emptyset$ then $\mu_0(A\udot B)=\mu_0(A)+\mu_0(B)$

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Follows immediately from definition (property 2)

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• Monotonic: if $A\subseteq B$ then $\mu_0(A)\le\mu_0(B)$

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TODO: Be bothered to write out

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• If $A\subseteq B$ and $\mu_0(A)<\infty$ then $\mu_0(B-A)=\mu_0(B)-\mu(A)$

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TODO: Be bothered, note the significance of the finite-ness of $A$ - see Extended real value

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• Strongly additive: $\mu_0(A\cup B)=\mu_0(A)+\mu_0(B)-\mu_0(A\cap B)$

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TODO: Be bothered

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• Subadditive: $\mu_0(A\cup B)\le\mu_0(A)+\mu_0(B)$

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TODO: Again - be bothered

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