A probability space is the measure-theoretic stage on which random objects live. The main point is that probabilities are assigned only to measurable events, and almost-sure statements are statements modulo null sets.
Events and probability measures
The axioms are the same as measure theory, normalized to total mass one.
Definition. A probability space is a triple \((\Omega,\mathcal F,P)\), where \(\mathcal F\) is a \(\sigma\)-field on \(\Omega\) and \(P:\mathcal F\to[0,1]\) is countably additive with \(P(\Omega)=1\). Events are elements of \(\mathcal F\). A property holds almost surely if it fails on a set \(N\in\mathcal F\) with \(P(N)=0\).
Proposition (probability calculus). For events \(A,B,A_n\), \[ \begin{aligned} P(A^c)&=1-P(A),\\ P(A\cup B)&=P(A)+P(B)-P(A\cap B),\\ P\!\left(\bigcup_{n\ge1}A_n\right)&\le\sum_{n\ge1}P(A_n). \end{aligned} \] If \(A\subset B\), then \(P(A)\le P(B)\) and \(P(B\setminus A)=P(B)-P(A)\).
Proof. The decompositions \[ \Omega=A\sqcup A^c,\qquad A\cup B=A\sqcup(B\setminus A),\qquad B=A\sqcup(B\setminus A) \] give the finite identities. For the union bound, disjointify: \[ B_1=A_1,\qquad B_n=A_n\setminus\bigcup_{k<n}A_k. \] Then \(\bigcup_nB_n=\bigcup_nA_n\) and \(B_n\subset A_n\), so countable additivity and monotonicity yield the inequality.
Example (canonical sample spaces depend on the question). For one coin toss, \(\Omega=\{0,1\}\) with its power set suffices. For an infinite sequence of tosses, the natural space is \[ \Omega=\{0,1\}^{\mathbb N}, \qquad \mathcal F=\sigma\{\omega:\omega_1=a_1,\ldots,\omega_k=a_k\}, \] the sigma-field generated by finite-coordinate cylinders. The event “the limiting frequency of heads is \(1/2\)” depends on infinitely many coordinates but is measurable because it is obtained from countable unions and intersections of cylinder-measurable frequency events.
Continuity of probability
Countable additivity is most often used through continuity along monotone set limits.
Theorem. If \(A_n\uparrow A\), then \(P(A_n)\uparrow P(A)\). If \(A_n\downarrow A\), then \(P(A_n)\downarrow P(A)\).
Proof. For \(A_n\uparrow A\), write \(B_1=A_1\) and \(B_n=A_n\setminus A_{n-1}\) for \(n\ge2\). The \(B_n\) are disjoint, \(A_n=\bigcup_{k\le n}B_k\), and \(A=\bigcup_{k\ge1}B_k\), so countable additivity gives \(P(A_n)=\sum_{k\le n}P(B_k)\to\sum_{k\ge1}P(B_k)=P(A)\). For \(A_n\downarrow A\), apply the increasing result to \(A_n^c\uparrow A^c\) and use \(P(A_n)=1-P(A_n^c)\).
Definition. The event that \(A_n\) occurs infinitely often and the event that it occurs eventually are \[ \limsup_nA_n=\bigcap_{m\ge1}\bigcup_{n\ge m}A_n, \qquad \liminf_nA_n=\bigcup_{m\ge1}\bigcap_{n\ge m}A_n. \]
Proposition. \[ P(\liminf_nA_n)\le\liminf_nP(A_n) \le\limsup_nP(A_n)\le P(\limsup_nA_n). \]
Proof. Put \(B_m=\bigcap_{n\ge m}A_n\uparrow\liminf A_n\). Since \(B_m\subset A_n\) for every \(n\ge m\), \[ P(B_m)\le\inf_{n\ge m}P(A_n). \] Let \(m\to\infty\). The upper inequality follows by applying the lower one to \(A_n^c\), or directly from \(C_m=\bigcup_{n\ge m}A_n\downarrow\limsup A_n\).
Null sets and completion
Almost-sure equality is harmless only when the measurable structure has enough null subsets.
Proposition. The completion of \((\Omega,\mathcal F,P)\) is obtained by adding every subset of every \(P\)-null set to \(\mathcal F\), and extending \(P\) by assigning those subsets probability zero.
Proof. Define \[ \overline{\mathcal F} =\{E:\text{there are }A,B\in\mathcal F,\ A\subset E\subset B,\ P(B\setminus A)=0\}. \] Equivalently, \(E\triangle A\) is contained in a measurable null set. This class is a sigma-field. Set \(\overline P(E)=P(A)\). Any two measurable representatives differ only by a null set, so the value is well-defined; replacing completed events by representatives proves countable additivity. Every subset of an \(\overline P\)-null event belongs to \(\overline{\mathcal F}\).
Remark. Completeness is convenient, but law-dependent. A Borel path space can support many probability laws, each with a different collection of null subsets; completing it too early can destroy a common measurable structure. Canonical stochastic-process constructions therefore often retain the Borel sigma-field and complete only after fixing the law.