The Pi-Lambda Criterion for Independence

Why a Generating-Class Test Is Useful

Independence of two sigma-algebras is defined by a factorization identity for every pair of events, one from each sigma-algebra. In applications, that is a large collection of tests: sigma-algebras can contain far more events than the classes that first describe the problem.

The pi-lambda criterion provides a rigorous way to reduce the work. If the identity holds on two suitable generating classes, then closure arguments extend it to the sigma-algebras those classes generate. The important point is that the classes must have the right structure; testing a few convenient events alone is not enough.

Pi-Systems and Lambda-Systems

Definition: A pi-system on a set $\Omega$ is a nonempty collection $\mathcal P$ of subsets of $\Omega$ that contains $\Omega$ and is closed under finite intersections. In symbols, for every $A,B\in\mathcal P$, $A\cap B\in\mathcal P$.

Definition: A lambda-system on $\Omega$ is a collection $\mathcal D$ of subsets of $\Omega$ that contains $\Omega$, is closed under complements relative to $\Omega$, and is closed under countable disjoint unions. Thus, if $A\in\mathcal D$, then $A^c\in\mathcal D$; and if $A_1,A_2,\ldots$ are pairwise disjoint members of $\mathcal D$, then $\bigcup_{n=1}^{\infty}A_n\in\mathcal D$.

A lambda-system need not initially be closed under arbitrary countable unions or intersections. One useful consequence of its defining properties is closure under nested differences: if $C,A\in\mathcal D$ and $C\subseteq A$, then $A\setminus C\in\mathcal D$. Indeed, $A^c$ and $C$ are disjoint members of $\mathcal D$, so $A^c\cup C\in\mathcal D$; taking its complement gives $A\setminus C$.

The Pi-Lambda Theorem

Theorem: If a pi-system $\mathcal P$ is contained in a lambda-system $\mathcal D$, then the sigma-algebra generated by $\mathcal P$ is contained in $\mathcal D$. Equivalently, the smallest lambda-system containing a pi-system $\mathcal P$ is the sigma-algebra $\sigma(\mathcal P)$.

Proof. Let $\mathcal L$ be the intersection of all lambda-systems containing $\mathcal P$. Intersections of lambda-systems are lambda-systems, so $\mathcal L$ is the smallest lambda-system containing $\mathcal P$. We will show that $\mathcal L$ is a sigma-algebra.

For a fixed $A\in\mathcal P$, define $\mathcal L_A=\{B\in\mathcal L:A\cap B\in\mathcal L\}$. This is a lambda-system. It contains $\Omega$, because $A\cap\Omega=A\in\mathcal L$. If $B\in\mathcal L_A$, then $A\cap B\in\mathcal L$, and $A\cap B\subseteq A$. The nested-difference property gives $A\setminus(A\cap B)=A\cap B^c\in\mathcal L$, so $B^c\in\mathcal L_A$. Finally, intersections with $A$ preserve disjointness and commute with unions, so $\mathcal L_A$ is closed under countable disjoint unions.

For every $B\in\mathcal P$, the intersection $A\cap B$ belongs to $\mathcal P$, since $\mathcal P$ is a pi-system. Thus $\mathcal P\subseteq\mathcal L_A$. By the minimality of $\mathcal L$, we have $\mathcal L\subseteq\mathcal L_A$. Consequently, $A\cap B\in\mathcal L$ whenever $A\in\mathcal P$ and $B\in\mathcal L$.

Now fix any $B\in\mathcal L$, and define $\mathcal E_B=\{A\in\mathcal L:A\cap B\in\mathcal L\}$. The same closure argument shows that $\mathcal E_B$ is a lambda-system. The conclusion of the preceding paragraph, together with symmetry of intersection, shows that $\mathcal P\subseteq\mathcal E_B$. Minimality again gives $\mathcal L\subseteq\mathcal E_B$. Hence $\mathcal L$ is closed under finite intersections.

A lambda-system closed under finite intersections is a sigma-algebra. It is closed under finite unions by complements and finite intersections. Given $A_1,A_2,\ldots\in\mathcal L$, form the disjoint sets $C_1=A_1$ and $C_n=A_n\setminus\bigcup_{k=1}^{n-1}A_k$ for $n\geq2$. Each $C_n$ belongs to $\mathcal L$, because the finite union being removed belongs to $\mathcal L$ and lambda-systems are closed under nested differences. The $C_n$ are pairwise disjoint and have union $\bigcup_n A_n$. Therefore $\bigcup_n A_n\in\mathcal L$. So $\mathcal L$ is a sigma-algebra containing $\mathcal P$, and $\sigma(\mathcal P)\subseteq\mathcal L$. Conversely, every sigma-algebra is a lambda-system, so $\mathcal L\subseteq\sigma(\mathcal P)$. This proves the theorem.

\mathcal L=\sigma(\mathcal P).

Independence on Generating Pi-Systems

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. Two sub-sigma-algebras $\mathcal A,\mathcal B\subseteq\mathcal F$ are independent if for every $A\in\mathcal A$ and $B\in\mathcal B$, $\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)$.

Theorem: Let $\mathcal P\subseteq\mathcal A$ and $\mathcal Q\subseteq\mathcal B$ be pi-systems containing $\Omega$, with $\sigma(\mathcal P)=\mathcal A$ and $\sigma(\mathcal Q)=\mathcal B$. If $$ \mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B) $$ for every $A\in\mathcal P$ and $B\in\mathcal Q$, then $\mathcal A$ and $\mathcal B$ are independent.

Proof. First fix $B\in\mathcal Q$ and consider the class $\mathcal D_B=\{A\in\mathcal A:\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)\}$. It is a lambda-system on $\Omega$. The whole space belongs to it because $\mathbb P(\Omega\cap B)=\mathbb P(B)=\mathbb P(\Omega)\mathbb P(B)$. If $A\in\mathcal D_B$, then \[ \mathbb P(A^c\cap B)=\mathbb P(B)-\mathbb P(A\cap B) =\mathbb P(B)-\mathbb P(A)\mathbb P(B) =\mathbb P(A^c)\mathbb P(B). \] Thus $A^c\in\mathcal D_B$. If the $A_n$ are pairwise disjoint members, countable additivity gives $\mathbb P((\bigcup_n A_n)\cap B)=\sum_n\mathbb P(A_n\cap B)=\mathbb P(B)\sum_n\mathbb P(A_n)=\mathbb P(B)\mathbb P(\bigcup_n A_n)$. So $\mathcal D_B$ is closed under countable disjoint unions. The assumed factorization gives $\mathcal P\subseteq\mathcal D_B$; by the pi-lambda theorem, $\mathcal A=\sigma(\mathcal P)\subseteq\mathcal D_B$.

Now fix any $A\in\mathcal A$ and define $\mathcal E_A=\{B\in\mathcal B:\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)\}$. The same calculations show that $\mathcal E_A$ is a lambda-system: the complement step subtracts $\mathbb P(A\cap B)$ from $\mathbb P(A)$, and the disjoint-union step follows from countable additivity. The first stage of the proof shows that every $B\in\mathcal Q$ belongs to $\mathcal E_A$. Applying the pi-lambda theorem once more gives $\mathcal B=\sigma(\mathcal Q)\subseteq\mathcal E_A$. Since this holds for every $A\in\mathcal A$, the two sigma-algebras are independent.

\square

Worked Examples

Worked Example: Two Coordinates on a Four-Point Space

Let $\Omega=\{(0,0),(0,1),(1,0),(1,1)\}$, with each point assigned probability $1/4$. Let $\mathcal P=\{\varnothing,\Omega,\{X=0\},\{X=1\}\}$ and $\mathcal Q=\{\varnothing,\Omega,\{Y=0\},\{Y=1\}\}$, where $X$ and $Y$ are the first and second coordinates. Each family is a pi-system: intersections of coordinate events are either the same event or empty, and intersections with $\Omega$ change nothing.

For $i,j\in\{0,1\}$, the event $\{X=i\}\cap\{Y=j\}$ is the single point $(i,j)$, so its probability is $1/4=(1/2)(1/2)$. If either event is empty, both sides of the factorization identity are zero; if either is $\Omega$, the identity reduces to $\mathbb P(C)=\mathbb P(C)$. Thus factorization holds for every pair in $\mathcal P\times\mathcal Q$. The theorem implies that $\sigma(X)$ and $\sigma(Y)$ are independent.

Worked Example: Rational Thresholds for Product Lebesgue Measure

Take $\Omega=[0,1]^2$ with product Lebesgue probability measure, and let $X(x,y)=x$ and $Y(x,y)=y$. Let $\mathcal P$ consist of $\varnothing,\Omega$, and the sets $\{X\leq r\}$ for rational $r\in[0,1]$; define $\mathcal Q$ using $\{Y\leq s\}$ instead. These are pi-systems because the intersection of two threshold events is the event at the smaller threshold. They generate the Borel sigma-algebras of the respective coordinates: differences give intervals with rational endpoints, and open subsets of $[0,1]$ are countable unions of such intervals.

For rational $r,s\in[0,1]$, the intersection is the rectangle $[0,r]\times[0,s]$. Its product measure and the marginal probabilities are \[ \mathbb P(X\leq r,Y\leq s)=rs,\qquad \mathbb P(X\leq r)=r,\qquad \mathbb P(Y\leq s)=s. \] Hence the required factorization holds. Empty sets and $\Omega$ also satisfy it directly. The pi-lambda criterion extends the identity to every pair of Borel events determined by the first and second coordinates.

Worked Example: Why the Pi-System Condition Matters

Let $\Omega=\{(x,y):x,y\in\{0,1\}\}$ have the uniform probability measure, and set $Z=X\mathbin{\mathrm{xor}}Y$, so $Z=0$ when $X=Y$ and $Z=1$ otherwise. Consider $\mathcal Q=\sigma(Z)$, and let $\mathcal P$ be the union of $\sigma(X)$ and $\sigma(Y)$. Every nontrivial event in either $\sigma(X)$ or $\sigma(Y)$ has probability $1/2$. Each event $\{Z=z\}$ contains exactly one point with $X=0$, one with $X=1$, one with $Y=0$, and one with $Y=1$. Thus, for any nontrivial coordinate event $A\in\mathcal P$ and $B\in\mathcal Q$, $\mathbb P(A\cap B)=1/4=\mathbb P(A)\mathbb P(B)$. Trivial events satisfy the identity as well.

Nevertheless, $\mathcal P$ is not a pi-system: $\{X=0\}$ and $\{Y=0\}$ belong to it, but their intersection $\{(0,0)\}$ does not. The sigma-algebra it generates contains $\{Z=0\}=\{(0,0),(1,1)\}$, since it contains the individual points obtained by intersecting $X$- and $Y$-coordinate events. For $A=B=\{Z=0\}$, factorization fails: \[ \mathbb P(A\cap B)=\frac12 \qquad\text{but}\qquad \mathbb P(A)\mathbb P(B)=\frac14. \] So testing on these non-pi-system classes does not justify extending the result to their generated sigma-algebras.

How to Use the Criterion

In a typical application, identify the two event classes that naturally arise from the problem, check that each contains $\Omega$ and is closed under finite intersections, and verify factorization for every pair from those classes. Then apply the theorem to extend the identity in two stages, first across one generated sigma-algebra and then across the other. This order is essential: the first application allows one event to range over its full sigma-algebra while the other remains in its generating class.

The main pitfall is to confuse a generating family with a pi-system. A family may generate a sigma-algebra without being closed under intersections. Factorization on such a family does not, by itself, provide the lambda-system closure argument needed for extension. Check the closure property before applying the criterion.

Check Your Understanding

Use the definitions and closure arguments above to answer the following questions.

What three closure properties define a lambda-system?
In the proof of the pi-lambda theorem, why does fixing $A\in\mathcal P$ produce a lambda-system $\mathcal L_A$?
Why is the whole space required to belong to each generating pi-system in the stated independence criterion?
In the independence proof, how does countable additivity establish closure of $\mathcal D_B$ under countable disjoint unions?
What specific intersection shows that the class $\mathcal P$ in the xor example is not a pi-system?