= Martingale construction of Lebesgue decomposition
The <Conditional Fatou lemma> gives $X_n\geq\mathbb E[X_\infty\mid\mathcal F_n]$ for the <dyadic density martingale>. Integration first on dyadic sets, then extension by the <Monotone class theorem>, makes $\nu=\mu-X_\infty\lambda$ a nonnegative <measure>. Its finite-level density is $Y_n=X_n-\mathbb E[X_\infty\mid\mathcal F_n]$, which tends to zero almost everywhere for <Lebesgue measure>. The <density level sets separate a singular measure> argument shows that this zero-limit set has $\nu$-measure zero. Thus $\nu$ and $\lambda$ are <mutually singular measures>, obtaining the decomposition without assuming it to identify the <martingale> limit.
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