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. Suppose that, for every x ∈ Rn, there exists a constant C = C(x) such that
\left|
| \int | |
| Rn |
D\varphi(y)(x)d\mu(y)\right|\leqC(x)\|\varphi\|infty
for every C∞ function φ : Rn → R with compact support. 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.
. Paul Malliavin. Stochastic calculus of variations and hypoelliptic operators. Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976). 195–263. Wiley. New York. 1978.