Lecture 14: The dyadic Plancherel theorem

Conditional expectation built from Radon-Nikodym and identified with the dyadic averages, then $L^2$ as an orthogonal sum over scales: the martingale differences are mutually orthogonal and $\|f\|_{L^2}^2 = \sum_k \|\mathbb{D}_k f\|_{L^2}^2$.