Suppose we want to assign sizes to uncertain events, but we also want those sizes to behave like honest mathematics. If an event is split into disjoint alternatives, the size of the whole should be the sum of the sizes of the alternatives. If an event is certain, its size should be $1$. If the sample space is continuous, such as the interval $[0,1]$, the same language should still describe events like $[0,1/2]$, countable unions of intervals, and limiting events that arise from repeated observations. A probability measure is the device that makes all of this possible.

The first obstruction is that not every subset of a sample space can always be assigned a consistent probability. In finite probability this issue is hidden, because every subset can be listed and counted. On an uncountable space, the attempt to assign lengths to all subsets while preserving countable additivity conflicts with the set-theoretic structure of the real line. The modern solution is to separate the sample space from the collection of events whose probabilities are being assigned.

[example: Why Events Need a Sigma Algebra] Let $\Omega=[0,1]$. Start with the open subintervals of $[0,1]$, and close them under complements in $\Omega$ and countable unions; the resulting collection is the Borel sigma algebra $\mathcal{B}([0,1])$. Because a sigma algebra is also closed under countable intersections, intervals such as $[a,b]$ are Borel: for $0\le a\le b\le 1$, \begin{align*} [a,b]=(a,1]\cap[0,b) \end{align*} where $(a,1]=\Omega\setminus[0,a]$ and $[0,b]=\Omega\setminus(b,1]$, and these sets are obtained from open intervals by complements and countable unions. On $\mathcal{B}([0,1])$, the uniform probability measure is Lebesgue length restricted to $[0,1]$, so \begin{align*} P([a,b])=b-a \end{align*} whenever $0\le a\le b\le 1$; in particular, \begin{align*} P([0,1])=1-0=1. \end{align*} The restriction to a sigma algebra is not cosmetic. If one tried to assign a probability to every subset while keeping countable additivity and [translation invariance](/theorems/4911) modulo $1$, a Vitali-type obstruction appears. Choose one representative from each equivalence class of $[0,1)$ under $x\sim y$ when $x-y\in\mathbb{Q}$, and call the resulting set $V$. For rational $r\in[0,1)\cap\mathbb{Q}$, let \begin{align*} V+r=\{(v+r)\bmod 1:v\in V\}. \end{align*} If $(V+r)\cap(V+s)$ contained a point, then for some $v,w\in V$ we would have $(v+r)\bmod 1=(w+s)\bmod 1$, hence $v-w\in\mathbb{Q}$, so $v=w$ by the choice of one representative from each equivalence class, and then $r=s$. Thus the sets $V+r$ are pairwise disjoint. They also cover $[0,1)$, because every $x\in[0,1)$ is rationally equivalent to exactly one representative $v\in V$, so $x=(v+r)\bmod 1$ for some rational $r\in[0,1)$. If such a probability measure measured all subsets and were invariant under these translations, then every set $V+r$ would have the same value $P(V)$. Countable additivity would give \begin{align*} 1=P([0,1))=\sum_{r\in[0,1)\cap\mathbb{Q}}P(V+r)=\sum_{r\in[0,1)\cap\mathbb{Q}}P(V). \end{align*} If $P(V)=0$, the right side is $0$; if $P(V)>0$, the right side diverges to infinity. Both contradict $P([0,1))=1$. The sigma algebra therefore records exactly which events the model is prepared to measure while preserving the probability rules. [/example]

The example shows the central pattern: probability is not only a number attached to an outcome, but a countably additive set function attached to a chosen class of events. The rest of the page stays close to that measure-theoretic core; random variables, laws, independence, and products are mentioned only as downstream uses of probability measures rather than developed as separate topics.

## Definition

The primary object is a normalized measure on a chosen family of events. The normalization $P(\Omega)=1$ marks the whole sample space as certain, while countable additivity says that disjoint alternatives contribute their probabilities without overlap. This is the point where ordinary intuition about finite counting becomes a definition that also works for intervals, limiting events, and infinite experiments.

[definition: Probability Measure] Let $\Omega$ be a set and let $\mathcal{F}$ be a sigma algebra on $\Omega$. A probability measure on $(\Omega,\mathcal{F})$ is a function \begin{align*} P:\mathcal{F} &\to [0,1] \end{align*} such that: \begin{align*} P(\Omega)&=1. \end{align*} \begin{align*} P\left(\bigcup_{n=1}^{\infty} A_n\right)&=\sum_{n=1}^{\infty} P(A_n) \end{align*} whenever $A_1,A_2,\ldots\in\mathcal{F}$ are pairwise disjoint. [/definition]

The notation $(\Omega,\mathcal{F})$ deliberately separates outcomes from events. The sample space $\Omega$ holds possible outcomes, but the probability measure is only required to evaluate sets in $\mathcal{F}$. To make that phrase precise, we isolate the event structure that the definition assumes.

## Event Structures

Before probability can be computed, we need a space of events stable under the operations that probability questions use. Complements model negation, countable unions model repeated alternatives, and countable intersections follow from the other two. This is why probability begins with a sigma algebra rather than with a bare collection of subsets.

[definition: Sigma Algebra] Let $\Omega$ be a set. A sigma algebra on $\Omega$ is a collection $\mathcal{F}\subset \mathcal{P}(\Omega)$ such that: $\Omega \in \mathcal{F}$. If $A\in\mathcal{F}$, then $\Omega\setminus A\in\mathcal{F}$. If $A_1,A_2,\ldots\in\mathcal{F}$, then $\bigcup_{n=1}^{\infty} A_n\in\mathcal{F}$. [/definition]

A sigma algebra is the bookkeeping device that makes limiting events legitimate, but the sigma algebra alone is not yet the object on which a probability measure is defined. We need a name for the sample space together with its admissible events, because the same underlying set can carry different event structures and therefore support different probability models. That paired object is the measurable space.

[definition: Measurable Space] A measurable space is a pair $(\Omega,\mathcal{F})$, where $\Omega$ is a set and $\mathcal{F}$ is a sigma algebra on $\Omega$. [/definition]

The elements of $\mathcal{F}$ are called measurable sets; in probability they are usually called events. With this supporting language in place, the definition of probability measure says exactly that events receive normalized, countably additive sizes.

Once probabilities are assigned only through normalization and countable additivity, even familiar manipulations such as passing to complements or taking limits of increasing events need justification. Without such rules, a calculation with events could silently use properties that have not been proved from the axioms. The following result collects the basic consequences that make probability measures usable in ordinary event calculations: monotonicity, complement formulas, finite inclusion-exclusion, and continuity along increasing or decreasing event sequences.

[quotetheorem:1106]

The last two identities are continuity statements. They explain why probability measures are compatible with limiting events, not only with finite Boolean combinations. That compatibility is what makes measure-theoretic probability suitable for sequences of random variables.

## Finite and Countable Models

Finite probability spaces reveal the definition without analytic complications. The event sigma algebra can be the full power set, and the probability measure is completely determined by the mass assigned to each point. To connect the measure-theoretic definition with the familiar act of assigning probabilities to individual outcomes, we first name the function that records those pointwise masses.

[definition: Probability Mass Function] Let $\Omega$ be a finite or [countable set](/page/Countable%20Set). A [probability mass function](/page/Probability%20Mass%20Function) on $\Omega$ is a function $p:\Omega\to[0,1]$ such that \begin{align*} \sum_{\omega\in\Omega}p(\omega)&=1. \end{align*} [/definition]