Churchill–Bernstein equation

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 Rⁿ. Suppose that, for every x ∈ 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^∞ 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.

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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