Product Probability Spaces and Joint Laws

From Two Probability Spaces to One

Two probability distributions can be combined in a canonical way: place their outcomes in a Cartesian product and assign each measurable rectangle the product of its two probabilities. The resulting probability space is useful in its own right, because its coordinate maps form a random pair with precisely that product law.

The word canonical matters. This construction specifies one joint distribution from two given probability measures. It does not imply that every pair of random elements with those individual distributions has that joint distribution. The sample space and the probability measure on it both matter.

The Product Measurable Space

A measurable rectangle records a restriction on each coordinate separately. The product sigma-algebra is the smallest sigma-algebra on \(E\times F\) that contains all such rectangles. In general, the definition concerns this generated sigma-algebra; it does not require that every subset of \(E\times F\) be measurable.

Suppose \(\mu\) is a probability measure on \((E,\mathcal E)\) and \(\nu\) is a probability measure on \((F,\mathcal F)\). The product-measure construction for sigma-finite measures, established in the earlier material on product measures, applies here: probability measures are finite and therefore sigma-finite.

The product-measure theorem guarantees existence, and uniqueness from measurable rectangles guarantees that these values determine the measure on the whole product sigma-algebra. Taking \(A=E\) and \(B=F\) gives \[ (\mu\otimes\nu)(E\times F)=\mu(E)\nu(F)=1, \] so the product measure is a probability measure.

The Canonical Coordinate Pair

On the product space itself, the two coordinate maps are defined by \[ \pi_E(x,y)=x \qquad\text{and}\qquad \pi_F(x,y)=y. \] Together they form the map \(\pi:E\times F\to E\times F\) given by \(\pi(x,y)=(x,y)\). More generally, the point of considering the coordinate maps separately is that they are measurable random elements on the product probability space.

Proof. For \(A\in\mathcal E\), the inverse image \(\pi_E^{-1}(A)\) is \(A\times F\), which is in \(\mathcal E\otimes\mathcal F\). Thus \(\pi_E\) is measurable. For \(B\in\mathcal F\), \(\pi_F^{-1}(B)=E\times B\), so \(\pi_F\) is measurable as well.

To check measurability of the pair map, take any generating rectangle \(A\times B\). Then \[ \pi^{-1}(A\times B)=A\times B\in\mathcal E\otimes\mathcal F. \] The collection of subsets of \(E\times F\) whose inverse images under \(\pi\) are measurable is a sigma-algebra. Since it contains every generating rectangle, it contains \(\mathcal E\otimes\mathcal F\). Hence \(\pi\) is measurable. \(\square\)

Proof. The law of a measurable map \(Z\) from a probability space with measure \(P\) into a measurable space is the measure \(P\circ Z^{-1}\), defined by \(C\mapsto P(Z\in C)\) for measurable \(C\). For the coordinate pair, its law assigns to each \(C\in\mathcal E\otimes\mathcal F\) the value \[ (\mu\otimes\nu)\bigl(\pi^{-1}(C)\bigr). \] Since \(\pi(x,y)=(x,y)\), its inverse image of \(C\) is \(C\) itself. Thus the displayed value is \((\mu\otimes\nu)(C)\) for every measurable \(C\), proving the claim. Equivalently, on every rectangle \(A\times B\) the law assigns \(\mu(A)\nu(B)\), and uniqueness from rectangles identifies it as the product measure. \(\square\)

Worked Example: A Product on Two Finite Spaces

Joint Laws on a Common Probability Space

Now let \((\Omega,\mathcal A,P)\) be any probability space, and let \(X:\Omega\to E\) and \(Y:\Omega\to F\) be measurable maps. Their joint map is \[ (X,Y):\Omega\to E\times F,\qquad (X,Y)(\omega)=(X(\omega),Y(\omega)). \] For \(A\in\mathcal E\) and \(B\in\mathcal F\), its inverse image of a rectangle is \[ (X,Y)^{-1}(A\times B)=X^{-1}(A)\cap Y^{-1}(B), \] which belongs to \(\mathcal A\). Since rectangles generate \(\mathcal E\otimes\mathcal F\), the same sigma-algebra argument used above shows that \((X,Y)\) is measurable.

This definition allows joint laws other than product measures. A joint law records how the two coordinates occur together on the given probability space. It equals the canonical product measure only when the probabilities it assigns to measurable sets agree with that product measure. The next examples show why the distinction cannot be ignored.

What the Product Construction Does—and Does Not—Say

The product construction begins with two probability measures and produces a particular probability measure on the product measurable space. On its canonical probability space, the coordinate pair has exactly that constructed law. This statement is guaranteed by the rectangle formula and the uniqueness of the product measure.

It is a different question to start with two random elements defined on some other probability space. Their joint law is determined by the probability of each measurable set of paired outcomes, not by the product construction alone. The finite and continuous examples show that a pair may concentrate on a small subset of the product space even when the canonical product measure does not. Thus one should not infer that an arbitrary joint law equals a product measure simply because the same coordinate spaces, or the same individual distributions, are involved.

A useful way to keep the constructions distinct is to ask: Which probability space carries the pair, and which measure is being used to evaluate its events? For the canonical pair, the space is \(E\times F\) and the measure is \(\mu\otimes\nu\). For a general pair, the underlying space is \(\Omega\), and its joint law is obtained by mapping \(P\) through \((X,Y)\).