Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 7 / a

The probability measure $\mathbb{P}$ maps the sigma-algebra $\mathcal{F}$ of events into $[0,1]$. Its probability axioms are nonnegativity, normalization $\mathbb{P}(\Omega )=1$, and countable additivity: for every pairwise disjoint sequence of events $A_1,A_2,\dots$,The sample space is $\Omega$, and $\mathcal{F}$ specifies which subsets are events. Requiring countable additivity, rather than only its finite version, is part of the probability space definition.

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