In mathematics — specifically, in measure theory — Malliavin's absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (smoothness) theorems of the Malliavin calculus. Malliavin's lemma gives a sufficient condition for a finite Borel measure to be absolutely continuous with respect to Lebesgue measure.
Let μ be a finite Borel measure on n-dimensional Euclidean space R<sup>n</sup>. Suppose that, for every x ∈ R<sup>n</sup>, there exists a constant C = C(x) such that
for every C<sup>∞</sup> function à: R<sup>n</sup> → R with compact support. Then ü is absolutely continuous with respect to n-dimensional Lebesgue measure û<sup>n</sup> on R<sup>n</sup>. In the above, DÃÂ(y) denotes the Fréchet derivative of φ at y and ||ÃÂ||<sub>∞</sub> denotes the supremum norm of ÃÂ.