= Solution
The <probability axioms> first give $\mathbb P(\varnothing)=0$: apply <countable additivity> to $\Omega,\varnothing,\varnothing,\ldots$ and use normalization and nonnegativity. If $A\subseteq B$, then $B=A\mathbin{\dot\cup}(B\setminus A)$ is a disjoint <union> of events. Finite additivity, as a special case of <countable additivity>, gives
$$
\mathbb P(B)=\mathbb P(A)+\mathbb P(B\setminus A)\geq\mathbb P(A).
$$
Thus \b[a <probability measure> is monotone with respect to <set inclusion>]. The difference event is measurable because the events belong to a <sigma-algebra>.
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