Past exam of the probability, measure and martingales course of the University of Oxford / 2021 / 3 / a

For each integer $m\ge 1$, letIf $D_m$ had $m+1$ distinct members, finite additivity on their disjoint singleton Borel sets would give total measure at least $(m+1)/m>1$, contradicting $\mu (\mathbb{R})=1$. Thus $|D_m|\le m$.

Every positive singleton measure is at least $1/m$ for some sufficiently large $m$. Consequently the set of point atoms of a measure isA countable union of finite sets is at most countable. This proves the claim, including the possibility that the set is empty. The same argument gives the countability of point atoms of a finite measure, with $|D_m|\le m\mu (\mathbb{R})$ in place of $m$.

The product measure $\mu \otimes \nu$ is the unique probability measure on the sigma-algebra generated by measurable rectangles for whichHere the printed product notation $\mathcal{B}(\mathbb{R})\times \mathcal{B}(\mathbb{R})$ denotes the generated product sigma-algebra, conventionally written $\mathcal{B}(\mathbb{R})\otimes \mathcal{B}(\mathbb{R})$; it is not just the collection of rectangles. On ${\mathbb{R}}^2$ it equals $\mathcal{B}({\mathbb{R}}^2)$: coordinate projections show that rectangles are Borel sets, while open rectangles with rational endpoints give every open set as a countable union and hence generate the Borel sigma-algebra.

Existence is the product measure theorem for finite measures, which are in particular sigma-finite measures. Uniqueness follows because the rectangles form a pi-system generating the product sigma-algebra, and two finite measures agreeing there agree everywhere by the Dynkin lemma.^[ref]

Consider the Borel setIts indicator function is nonnegative and measurable, so the Tonelli theorem applies to the product measure, without needing any separate absolute-integrability argument. At a fixed $y\in (a,b]$, its section in the first coordinate is $(a,y]$; outside $(a,b]$ that section is empty. ThereforeThe last equality uses the definition of a cumulative distribution function on the half-closed interval. It retains all point atoms of a measure on the diagonal $x=y$, excludes a possible point atom of a measure at $x=a$, and includes $y=b$, exactly as required by the set $D$. If $a=b$, both sides vanish. This verifies the identity even when the probability measures are not non-atomic measures.^[ref]