Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / A / 1

A pi-system is a collection of subsets closed under binary intersections: $A,B\in \mathcal{I}$ implies $A\cap B\in \mathcal{I}$. Induction then gives closure under every nonempty finite intersection. A pi-system need not contain complements or unions.

The sigma-algebra generated by a collection is the smallest sigma-algebra containing that collection:The family being intersected is nonempty, since it contains the power set of $\Omega$. Intersections of sigma-algebras are sigma-algebras, so this definition has the asserted minimality.

The set function is countably additive if, whenever $A_1,A_2,\dots \in \mathcal{I}$ are pairwise disjoint and their union $A$ also belongs to $\mathcal{I}$,The equality is in the extended nonnegative reals. To be a measure one also imposes $\mu (\varnothing )=0$ and an appropriate domain such as a sigma-algebra.

The intersection of two nonempty members iswhich belongs to the collection. Any intersection involving the empty set is empty and is included too. Hence $\pi (\mathbb{R})$ is a pi-system.

Each half-line $(-\infty ,x]$ is a closed Borel set, and the empty set is Borel. Thus $\sigma (\pi (\mathbb{R}))\subseteq \mathcal{B}(\mathbb{R})$.

Conversely,The generated sigma-algebra contains both sets on the right and is closed under their difference, so it contains every open interval. By the given definition of the Borel sigma-algebra, the reverse inclusion follows. Therefore

Two sigma-algebras are independent if every event in one is independent of every event in the other:The same formula defines independence of the generating collections when restricted to their members.

Independence of the full sigma-algebras immediately gives independence of their generating pi-systems. For the converse, fix $H\in {\mathcal{H}}_0$. On $\mathcal{G}$, define finite measuresThey have the same total mass $\mathbb{P}(H)$ and agree on ${\mathcal{G}}_0$ by assumption. The supplied measure-uniqueness theorem makes them equal on $\mathcal{G}$. Thus the independence formula holds for every $G\in \mathcal{G}$ and every $H\in {\mathcal{H}}_0$.

Now fix such an arbitrary $G$. The measures $H\mapsto \mathbb{P}(G\cap H)$ and $H\mapsto \mathbb{P}(G)\mathbb{P}(H)$ on $\mathcal{H}$ have equal total mass and agree on ${\mathcal{H}}_0$. A second application gives equality on all of $\mathcal{H}$. This proves independence extended from generating pi-systems.

For each $i$, the collection of events $\{X_i\le x\}$, together with the empty event, is a pi-system. Its generated sigma-algebra is $\sigma (X_i)$, because the half-lines generate the Borel sigma-algebra and inverse images preserve intersections and the sigma-algebra operations.

The displayed probability identity says precisely that these two pi-systems are independent. Part (ii) makes $\sigma (X_1)$ and $\sigma (X_2)$ independent, which is the definition of independent random variables. Conversely independence of the random variables applies to the half-line events and gives the identity. Thus the joint cumulative distribution function factors exactly when the variables are independent.