# 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.

## Statement of the lemma

Let *μ* be a finite Borel measure on *n*-dimensional Euclidean space **R**^{n}. Suppose that, for every *x* ∈ **R**^{n}, there exists a constant *C* = *C*(*x*) such that

for every *C*^{∞} function *φ* : **R**^{n} → **R** with compact support. Then *μ* is absolutely continuous with respect to *n*-dimensional Lebesgue measure *λ*^{n} on **R**^{n}. In the above, D*φ*(*y*) denotes the Fréchet derivative of *φ* at *y* and ||*φ*||_{∞} denotes the supremum norm of *φ*.

## References

- Bell, Denis R. (2006).
*The Malliavin calculus*. Mineola, NY: Dover Publications Inc. pp. x+113. ISBN 0-486-44994-7. MR 2250060 (See section 1.3) - Malliavin, Paul (1978). "Stochastic calculus of variations and hypoelliptic operators".
*Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976)*. New York: Wiley. pp. 195–263. MR 536013