Probability Spaces, Light Version

A probability space is the mathematical structure that says which outcomes are possible, which collections of outcomes count as events, and how probabilities are assigned to those events.

Quantum mechanics does not remove the need for this language. The Born rule produces ordinary probabilities once a state and measurement have been specified. What is special is how those probabilities are generated, and which measurements can or cannot be treated as a single joint classical experiment.

This page gives only the vocabulary needed for quantum-mechanics pages. It is not a full measure-theory course.

Probability Space

A probability space is a triple

$$(\Omega,\mathcal F,\mathbb P).$$

The pieces are:

• $\Omega$: the sample space, or set of possible outcomes;
• $\mathcal F$: the collection of events to which probabilities can be assigned;
• $\mathbb P$: the probability measure assigning numbers to events.

An event is a set of outcomes. If $A\in\mathcal F$, then

$$\mathbb P(A)$$

is the probability that the outcome lies in $A$.

Events

Events can be combined with set operations:

• $A\cup B$: $A$ or $B$ occurs;
• $A\cap B$: both $A$ and $B$ occur;
• $A^c$: $A$ does not occur.

Two events are disjoint, or mutually exclusive, if

$$A\cap B=\emptyset.$$

For a finite spin measurement with outcomes $+$ and $-$, one can take

$$\Omega=\{+,-\}.$$

Then examples of events are

$$\emptyset, \qquad \{+\}, \qquad \{-\}, \qquad \{+,-\}.$$

The event $\{+\}$ means “the plus outcome occurs.” The event $\{+,-\}$ is certain because it contains all possible outcomes in this experiment.

Why Events Need a Collection

For finite or countable sample spaces, one often takes $\mathcal F$ to be all subsets of $\Omega$. For continuous sample spaces such as $\Omega=\mathbb R$, the event collection is usually the Borel sets: intervals and the sets built from intervals by countable unions, intersections, and complements.

The point of $\mathcal F$ is to say which sets are measurable. In elementary quantum mechanics, this usually stays invisible because intervals, finite unions of intervals, and ordinary regions in space are measurable. The vocabulary matters when one writes statements such as

$$\mathbb P(X\in\Delta),$$

where $\Delta$ is an allowed measurable set of values.

Probability Measure

A probability measure satisfies:

$$0\le\mathbb P(A)\le1,$$

for every event $A$,

$$\mathbb P(\Omega)=1,$$

and, for disjoint events $A_1,A_2,\ldots$,

$$\mathbb P \left( \bigcup_{n=1}^{\infty}A_n \right) = \sum_{n=1}^{\infty} \mathbb P(A_n).$$

The last property is countable additivity. In finite examples it reduces to the familiar rule that probabilities of mutually exclusive alternatives add.

From these rules one gets

$$\mathbb P(A^c) = 1-\mathbb P(A),$$

and, for any two events,

$$\mathbb P(A\cup B) = \mathbb P(A)+\mathbb P(B)-\mathbb P(A\cap B).$$

Random Variables

A random variable is a function from outcomes to values:

$$X:\Omega\to\mathbb R.$$

For example, if $\Omega$ is the set of die outcomes,

$$\Omega=\{1,2,3,4,5,6\},$$

then $X(\omega)=\omega$ is the face value. A different random variable could be

$$Y(\omega) = \begin{cases} 1, & \omega\ \text{is even},\\ 0, & \omega\ \text{is odd}. \end{cases}$$

In measure-theoretic language, a random variable must be measurable. The practical meaning is that events such as

$$\{\,\omega:X(\omega)\in\Delta\,\}$$

must be events in $\mathcal F$ whenever $\Delta$ is an allowed set of values.

The dedicated page is Random Variables.

Distribution of a Random Variable

The distribution of $X$ is the probability measure it induces on its values:

$$\mathbb P_X(\Delta) = \mathbb P(X\in\Delta).$$

For a discrete random variable, this is often recorded as probabilities $p_i$ for values $x_i$:

$$\mathbb P(X=x_i)=p_i.$$

For a continuous random variable, it may be described by a density $f(x)$:

$$\mathbb P(X\in[a,b]) = \int_a^b f(x)\,dx.$$

The density is not itself a probability. It becomes a probability after integration over an event. The dedicated density page is Probability Densities.

Expectation Preview

The expectation value is the average of a random variable with respect to its distribution. In the discrete case,

$$\mathbb E[X] = \sum_i x_i p_i.$$

In the continuous-density case,

$$\mathbb E[X] = \int_{-\infty}^{\infty} x f(x)\,dx,$$

when the integral exists. The dedicated page is Expectation Values.

Quantum Measurements as Probability Spaces

A quantum state and a specified measurement define an ordinary probability distribution over that measurement’s outcomes.

For a finite projective measurement with projectors $P_a$ and a normalized state $\lvert\psi\rangle$, the sample space can be the set of outcome labels:

$$\Omega=\{a\}.$$

The Born probabilities are

$$\mathbb P(\{a\}) = \langle\psi\vert P_a\lvert\psi\rangle.$$

For a density operator $\rho$, the same probabilities are

$$\mathbb P(\{a\}) = \operatorname{Tr}(\rho P_a).$$

This is a classical probability measure on the outcome set, produced by quantum structure. The canonical physics page is Born Rule.

Continuous Quantum Outcomes

For position on the real line, the sample space for the measurement can be

$$\Omega=\mathbb R,$$

with events such as intervals or regions. If $\psi(x)$ is normalized, the probability of finding the particle in a region $\Delta$ is

$$\mathbb P(X\in\Delta) = \int_\Delta \lvert\psi(x)\rvert^2\,dx.$$

Here $\lvert\psi(x)\rvert^2$ is a probability density. It is not the probability of an exact point. The continuous-spectrum treatment is Born Rule for Continuous Spectra.

Not One Universal Classical Sample Space

In classical probability, it is often natural to imagine a single sample point carrying values of many random variables at once. Quantum mechanics is more careful. A state plus one measurement gives a probability space for that measurement. A different, incompatible measurement generally gives a different probability distribution, and there may be no single joint probability distribution for all observables that preserves the quantum predictions.

This does not mean quantum probabilities are not probabilities. It means the measurement context matters. The probability space is attached to the experiment being described. The fuller comparison is Classical Probability versus Quantum Probability.

Minimal Vocabulary

Useful terms:

• Outcome: one possible result of a specified experiment.
• Sample space: the set $\Omega$ of possible outcomes.
• Event: a measurable subset of $\Omega$.
• Probability measure: a rule $\mathbb P$ assigning probabilities to events.
• Random variable: a measurable function from outcomes to values.
• Distribution: the induced probability law of a random variable.
• Almost surely: true except on an event of probability zero.
• Support: the region where a distribution can assign probability.

The phrase “almost surely” is especially useful in continuous settings. A statement can fail at isolated points and still hold with probability one if those points have probability zero.

Common Mistakes

• Confusing an outcome with an event.
• Treating a probability density as a probability.
• Forgetting to specify the measurement before assigning quantum probabilities.
• Assuming every subset of a continuous sample space is automatically measurable.
• Treating incompatible quantum observables as ordinary random variables on one hidden sample space without additional assumptions.
• Reading probability-zero events in continuous distributions as logically impossible.

Cross-Links

• Random Variables
• Probability Densities
• Expectation Values
• Variance and Covariance
• Conditional Probability
• Bayes’ Rule
• Characteristic Functions
• Gaussian Distributions
• Entropy
• Classical Probability versus Quantum Probability
• Born Rule
• Born Rule for Continuous Spectra
• Quantum Expectation Values
• Density Operators

References

• A. N. Kolmogorov, Foundations of the Theory of Probability, 2nd English ed., Chelsea, 1956.
• W. Feller, An Introduction to Probability Theory and Its Applications, Volume I, 3rd ed., Wiley, 1968.
• P. Billingsley, Probability and Measure, 3rd ed., Wiley, 1995.
• R. Durrett, Probability: Theory and Examples, 5th ed., Cambridge University Press, 2019.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.

Exercises

1. Let $\Omega=\{H,T\}$ for a coin toss. List all events if every subset is measurable.

Solution

The events are

$$\emptyset, \qquad \{H\}, \qquad \{T\}, \qquad \{H,T\}.$$

2. If $A$ and $B$ are disjoint events with $\mathbb P(A)=0.2$ and $\mathbb P(B)=0.3$, what is $\mathbb P(A\cup B)$?

Solution

Since $A$ and $B$ are disjoint, additivity gives

$$\mathbb P(A\cup B) = \mathbb P(A)+\mathbb P(B) = 0.5.$$

3. For a fair die, let $Y=1$ for even outcomes and $Y=0$ for odd outcomes. Find the distribution of $Y$.

Solution

There are three even outcomes and three odd outcomes. Therefore

$$\mathbb P(Y=1)=\frac{3}{6}=\frac12, \qquad \mathbb P(Y=0)=\frac{3}{6}=\frac12.$$

4. A two-outcome projective measurement has projectors $P_0$ and $P_1$ with $P_0+P_1=I$. For a normalized state $\lvert\psi\rangle$, write the probability space for this measurement.

Solution

Take

$$\Omega=\{0,1\},$$

with all subsets measurable. Define

$$\mathbb P(\{0\}) = \langle\psi\vert P_0\lvert\psi\rangle, \qquad \mathbb P(\{1\}) = \langle\psi\vert P_1\lvert\psi\rangle.$$

Since $P_0+P_1=I$ and $\lVert\psi\rVert=1$,

$$\mathbb P(\{0\})+\mathbb P(\{1\}) = \langle\psi\vert I\lvert\psi\rangle = 1.$$

5. If a normalized position wavefunction has density $f(x)=\lvert\psi(x)\rvert^2$, what is the probability of finding the particle in $[a,b]$?

Solution

The event is the interval $[a,b]$, and the probability is

$$\mathbb P(X\in[a,b]) = \int_a^b \lvert\psi(x)\rvert^2\,dx.$$

The value $f(x)$ at a point is a density, not a point probability.