Result in measure theory
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
formula_1
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".