Dyadic Sigma-Algebras on a Compact Interval and Conditional Expectation as Dyadic Averaging

Let $T$ be a real number with $T>0$. Adopt the notation $\mathcal{B}_{[0,T]}$ and $\lambda_{[0,T]}$ of the restricted Lebesgue measure space on a compact interval, and let $P_{T}(E)=T^{-1}\lambda_{[0,T]}(E)$ for $E\in\mathcal{B}_{[0,T]}$, so that $([0,T],\mathcal{B}_{[0,T]},P_{T})$ is a probability space by claim 1 of that lemma. Write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$, and $\mathbb{E}$ for the expectation on this probability space.

For a natural number $m\ge1$ and $p\in\{1,\dots,2^{m}\}$ define the level-$m$ dyadic atoms

$$I_{m,p}=\bigl[T(p-1)2^{-m},\,Tp\,2^{-m}\bigr)\ \ (p<2^{m}),\qquad I_{m,2^{m}}=\bigl[T(2^{m}-1)2^{-m},\,T\bigr],$$

and let $\mathcal{G}_{m}$ be the collection of all unions of level-$m$ dyadic atoms, the empty union being $\emptyset$. Then the following hold.

1. (The dyadic filtration.) For each $m\ge1$ the sets $I_{m,1},\dots,I_{m,2^{m}}$ are pairwise disjoint members of $\mathcal{B}_{[0,T]}$ whose union is $[0,T]$, and $P_{T}(I_{m,p})=2^{-m}$ for every $p$. The collection $\mathcal{G}_{m}$ is a $\sigma$-algebra on $[0,T]$ with $\mathcal{G}_{m}\subseteq\mathcal{B}_{[0,T]}$. Moreover each level-$m$ atom is the union of two level-$(m+1)$ atoms,

$$I_{m,p}=I_{m+1,2p-1}\cup I_{m+1,2p}\qquad(1\le p\le2^{m}),$$

and consequently $\mathcal{G}_{m}\subseteq\mathcal{G}_{m+1}$; more generally, for natural numbers $m'\ge m$ every level-$m'$ atom is contained in exactly one level-$m$ atom.

2. (The filtration generates the trace Borel $\sigma$-algebra.) The generated $\sigma$-algebra of the union satisfies

$$\sigma\Bigl(\bigcup_{m\ge1}\mathcal{G}_{m}\Bigr)=\mathcal{B}_{[0,T]} .$$

3. (Conditional expectation is dyadic averaging.) Let $X$ be a square-integrable random variable on $([0,T],\mathcal{B}_{[0,T]},P_{T})$ and let $m\ge1$. Then the function

$$A_{m}X=\sum_{p=1}^{2^{m}}2^{m}\,\mathbb{E}\bigl[X\mathbf{1}_{I_{m,p}}\bigr]\,\mathbf{1}_{I_{m,p}}$$

is a bounded square-integrable random variable and is a conditional expectation of $X$ given $\mathcal{G}_{m}$.