Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 33 / 1 / a / Solution

Use the filtered probability space $([0,1),\mathcal{B}([0,1)),({\mathcal{F}}_n),\lambda )$, with its half-open dyadic intervals. Complete the initial definition by putting $X_0=\mu ([0,1))$. This is the same cell-average formula at level zero, where the only cell is $[0,1)$.

The function $X_n$ is ${\mathcal{F}}_n$-measurable and nonnegative. Its expectation isEvery parent cell is the disjoint union of its two half-open child cells. On $D_k^n$, the conditional expectation of $X_{n+1}$ is its cell average:Thus $\mathbb{E}[X_{n+1}\mid {\mathcal{F}}_n]=X_n$, including at $n=0$. This proves that $X$ is a nonnegative martingale, specifically the dyadic density martingale. The half-open endpoints are essential to the disjoint partition for an arbitrary measure, which may have atoms.