Consequences of Kolmogorov's Axioms

What the Probability Axioms Imply

A probability measure is not merely a rule for assigning numbers between zero and one to events. Its axioms force relationships among those numbers. In particular, they determine the probability of a complement, give addition rules for disjoint events, and control the probability of a union even when its events overlap. These consequences are useful both for exact calculations and for bounding probabilities when the full structure of an event is not known.

Throughout, \((\Omega,\mathcal{F},\mathbb{P})\) is a probability space: \(\Omega\) is a sample space, \(\mathcal{F}\) is a sigma-algebra of subsets of \(\Omega\), and \(\mathbb{P}\) is a measure on \(\mathcal{F}\) satisfying \(\mathbb{P}(\Omega)=1\). The members of \(\mathcal{F}\) are called events. Thus \(\mathbb{P}\) is a finite measure, and in particular is sigma-finite. It need not be complete: a subset of an event of probability zero need not itself belong to \(\mathcal{F}\). All events below belong to \(\mathcal{F}\), and all equalities and inequalities are numerical statements about their probabilities.

Countable additivity applies to finite disjoint collections as well: append empty events after the last set. The first consequence needed is that the empty event has probability zero. From there, disjoint additivity gives the formulas for complements and for general pairs of events.

Complements, Additivity, and Bounds

Proof. Since \(\Omega\) and \(\varnothing\) are disjoint and their union is \(\Omega\), additivity and normalization give \(1=\mathbb{P}(\Omega)=\mathbb{P}(\Omega)+\mathbb{P}(\varnothing)=1+\mathbb{P}(\varnothing)\). Hence \(\mathbb{P}(\varnothing)=0\). The sets \(A\) and \(A^c\) are disjoint and have union \(\Omega\), so additivity gives \(1=\mathbb{P}(A)+\mathbb{P}(A^c)\), proving the complement formula. If \(A\subseteq B\), then \(B\) is the disjoint union of \(A\) and \(B\setminus A\). Therefore \(\mathbb{P}(B)=\mathbb{P}(A)+\mathbb{P}(B\setminus A)\ge\mathbb{P}(A)\), by nonnegativity. The disjoint-additivity assertion is the probability axiom, applied to \(A,B\), followed by empty events. For the general formula, decompose \(A\cup B\) into the disjoint sets \(A\) and \(B\setminus A\). Also decompose \(B\) into the disjoint sets \(A\cap B\) and \(B\setminus A\). Thus \(\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B\setminus A)\) and \(\mathbb{P}(B)=\mathbb{P}(A\cap B)+\mathbb{P}(B\setminus A)\). Substituting the second equality into the first proves the stated identity. \(\square\)

In particular, every event has probability between zero and one: nonnegativity gives the lower bound, and monotonicity applied to \(A\subseteq\Omega\) gives \(\mathbb{P}(A)\le\mathbb{P}(\Omega)=1\). The two-event identity also gives a lower bound for an intersection.

Proof. The intersection is a subset of each event, so monotonicity gives both upper bounds. Its probability is nonnegative. For the other lower bound, the union identity and \(\mathbb{P}(A\cup B)\le1\) imply \(\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)\le1\). Rearranging gives \(\mathbb{P}(A\cap B)\ge\mathbb{P}(A)+\mathbb{P}(B)-1\). Combining this with nonnegativity proves the result. \(\square\)

Union Bounds from Countable Additivity

When events overlap, their probabilities cannot generally be added to get the probability of their union. They can, however, be added to obtain an upper bound. This is the union bound, also called countable subadditivity for probability. Its proof turns overlapping events into disjoint pieces, so that the probability axiom can be applied directly.

Proof. Define \(D_1=A_1\), and for \(n\ge2\) define \(D_n=A_n\setminus\bigcup_{j=1}^{n-1}A_j\). These are events because a sigma-algebra is closed under finite unions and differences. If \(i<n\), then \(D_n\) excludes \(A_i\), while \(D_i\subseteq A_i\); hence \(D_i\cap D_n=\varnothing\). Every point in \(\bigcup_n A_n\) has a first index at which it belongs to an \(A_n\), so it belongs to the corresponding \(D_n\); conversely \(D_n\subseteq A_n\). Therefore \(\bigcup_nD_n=\bigcup_nA_n\). Countable additivity and monotonicity now give $$ \mathbb{P}\left(\bigcup_n A_n\right) =\sum_n\mathbb{P}(D_n) \le\sum_n\mathbb{P}(A_n), $$ because \(D_n\subseteq A_n\) for each \(n\). The finite version follows by applying the same argument to the finite list, or by appending empty events. \(\square\)

Sharp Bounds and a Common Mistake

The two-event lower bound can be attained: if two events are large enough that their probabilities sum to more than one, some overlap is unavoidable. The next example reaches exactly the lower bound. It also illustrates why bounds can be useful even when the event intersection is not known in advance.

Common mistake: adding probabilities of overlapping events as though they were disjoint. In the eight-outcome example, the incorrect calculation \(\mathbb{P}(A\cup B)=4/8+3/8=7/8\) counts outcomes \(3\) and \(4\) twice. The actual union has five outcomes, so its probability is \(5/8\). The correction is to subtract the overlap: \(4/8+3/8-2/8=5/8\). If the overlap is not known, the union bound \(7/8\) is still a valid upper bound, but it is not generally an equality.

These formulas require no independence assumption. They follow from the measure axioms and set relationships alone. In applications, the exact union identity is useful when the overlap can be computed; the union bound remains available when only the individual event probabilities are known. The bound may be loose, but it cannot underestimate the probability that at least one event occurs.