Dyadic conditional expectation martingale

First justify the finite integral operations locally. Augment every finite disjoint simple display by the complement with coefficient $0$. Intersections of two augmented displays partition the whole space and have equal coefficients on every nonempty cell, so finite additivity and $0\cdot (+\mathrm{\infty })=0$ prove representation independence. Common refinements give simple addition and monotonicity; scalar zero is direct and positive scalars are termwise. Supremum over simple minorants, increasing simple approximation and the sets $\{f_j\ge cs\}$, $0<c<1$, give monotone convergence and nonnegative additivity. Positive/negative and real/imaginary decompositions give the finite $L^1$ linearity and triangle estimate used below. The trace restriction of Lebesgue measure to $(0,1]$ has total mass one. The displayed half-open cells Half-open boxes in ${\mathbb{R}}^n$ and their volume are disjoint, exhaust the space and have probability $2^{-n}>0$. Their unions form a sigma-algebra because unions and complements select cells from a finite partition. Each cell at n is exactly the union of its children at n+1, so the sigma-algebras increase. Every cell integral of $f$ is finite. The displayed finite-valued $M_n$ is ${\mathcal{F}}_n$-measurable, and $E\mathrm{\mid }{M}_n\mathrm{\mid }={\sum }_j\mathrm{\mid }{\int }_{I_{n,j}}fdP\mathrm{\mid }\le {\sum }_j{\int }_{I_{n,j}}\mathrm{\mid }f\mathrm{\mid }dP=E\mathrm{\mid }f\mathrm{\mid }<\mathrm{\infty }$.

For $A={\bigcup }_{j\in J}{I}_{n,j}\in {\mathcal{F}}_n$, finite addition and the cell mass give ${\int }_A{M}_{n}dP={\sum }_{j\in J}{2}^n({\int }_{I_{n,j}}fdP)2^{-n}={\int }_{A}fdP$. Thus $M_n$ meets every defining conditional-expectation requirement. Apply [F6] to obtain the martingale. For $f={\mathbf{1}}_{(0,1\mathrm{/}2]}$, the first average is $M_0=1\mathrm{/}2$ and every finer cell lies entirely in $(0,1\mathrm{/}2]$ or its complement, so $M_n=f$ for $n\ge 1$. In particular $E[M_1\mid {\mathcal{F}}_0]=1\mathrm{/}2$. AC supplies the Lebesgue construction and CE existence; the explicit finite averages require no version selection.