mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

— specifically, in

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

— Malliavin's absolute continuity lemma is a result due to the

French French may refer to: * Something of, from, or related to France ** French language, which originated in France ** French people, a nation and ethnic group ** French cuisine, cooking traditions and practices Arts and media * The French (band), ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Paul Malliavin Paul Malliavin (; September 10, 1925 – June 3, 2010) was a French mathematician who made important contributions to harmonic analysis and stochastic analysis. He is known for the Malliavin calculus, an infinite dimensional calculus for func ...

that plays a foundational rôle in the regularity (

smoothness In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

)

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

s of the

Malliavin calculus In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...

. Malliavin's lemma gives a sufficient condition for a

finite Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...

to be

absolutely continuous In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship betwe ...

with respect to

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

Statement of the lemma

Let ''μ'' be a finite Borel measure on ''n''-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

R^''n''. Suppose that, for every ''x'' ∈ R^''n'', there exists a constant ''C'' = ''C''(''x'') such that :

\left,  \int_ \mathrm \varphi (y) (x) \, \mathrm \mu(y) \ \leq C(x) \,  \varphi \, _

for every ''C''^∞ function ''φ'' : R^''n'' → R with

compact support In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...

. 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 In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued f ...

of ''φ'' at ''y'' and , , ''φ'', , _∞ denotes the

supremum norm In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when t ...

of ''φ''.

References

* (See section 1.3) * {{MathSciNet, id=536013 Lemmas in mathematical analysis Measure theory Malliavin calculus