Malliavin's absolute continuity lemma

In mathematics &mdash; specifically, in measure theory &mdash; 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.

Statement of the lemma
Let &mu; be a finite Borel measure on n-dimensional Euclidean space Rn. Suppose that, for every x &isin; Rn, there exists a constant C = C(x) such that


 * $$\left| \int_{\mathbf{R}^{n}} \mathrm{D} \varphi (y) (x) \, \mathrm{d} \mu(y) \right| \leq C(x) \| \varphi \|_{\infty}$$

for every C&infin; function φ : Rn &rarr; 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 &phi; at y and ||φ||&infin; denotes the supremum norm of φ.