Malliavin's absolute continuity lemma

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 Rn.

Then μ is absolutely continuous with respect to n-dimensional Lebesgue measure λn on Rn.

In the above, Dφ(y) denotes the Fréchet derivative of φ at y and ||φ||∞ denotes the supremum norm of φ.