Differentiation of integrals

Short description: Problem in mathematics

In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks for what functions f : X → R does

[math]\displaystyle{ \lim_{r \to 0} \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu(y) = f(x) }[/math]

for all (or at least μ-almost all) x ∈ X? (Here, as in the rest of the article, Br(x) denotes the open ball in X with d-radius r and centre x.) This is a natural question to ask, especially in view of the heuristic construction of the Riemann integral, in which it is almost implicit that f(x) is a "good representative" for the values of f near x.

## Contents

* 1 Theorems on the differentiation of integrals
* 1.1 Lebesgue measure
* 1.2 Borel measures on Rn
* 1.3 Gaussian measures
* 2 See also
* 3 References

## Theorems on the differentiation of integrals

### Lebesgue measure

One result on the differentiation of integrals is the Lebesgue differentiation theorem, as proved by Henri Lebesgue in 1910. Consider n-dimensional Lebesgue measure λn on n-dimensional Euclidean space Rn. Then, for any locally integrable function f : Rn → R, one has

[math]\displaystyle{ \lim_{r \to 0} \frac1{\lambda^{n} \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \lambda^{n} (y) = f(x) }[/math]

for λn-almost all points x ∈ Rn. It is important to note, however, that the measure zero set of "bad" points depends on the function f.

### Borel measures on Rn

The result for Lebesgue measure turns out to be a special case of the following result, which is based on the Besicovitch covering theorem: if μ is any locally finite Borel measure on Rn and f : Rn → R is locally integrable with respect to μ, then

[math]\displaystyle{ \lim_{r \to 0} \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu (y) = f(x) }[/math]

for μ-almost all points x ∈ Rn.

### Gaussian measures

The problem of the differentiation of integrals is much harder in an infinite-dimensional setting. Consider a separable Hilbert space (H, ⟨ , ⟩) equipped with a Gaussian measure γ. As stated in the article on the Vitali covering theorem, the Vitali covering theorem fails for Gaussian measures on infinite-dimensional Hilbert spaces. Two results of David Preiss (1981 and 1983) show the kind of difficulties that one can expect to encounter in this setting:

* There is a Gaussian measure γ on a separable Hilbert space H and a Borel set M ⊆ H so that, for γ-almost all x ∈ H, [math]\displaystyle{ \lim_{r \to 0} \frac{\gamma \big( M \cap B_{r} (x) \big)}{\gamma \big( B_{r} (x) \big)} = 1. }[/math]
* There is a Gaussian measure γ on a separable Hilbert space H and a function f ∈ L1(H, γ; R) such that [math]\displaystyle{ \lim_{r \to 0} \inf \left\\{ \left. \frac1{\gamma \big( B_{s} (x) \big)} \int_{B_{s} (x)} f(y) \, \mathrm{d} \gamma(y) \right| x \in H, 0 \lt s \lt r \right\\} = + \infty. }[/math]

However, there is some hope if one has good control over the covariance of γ. Let the covariance operator of γ be S : H → H given by

[math]\displaystyle{ \langle Sx, y \rangle = \int_{H} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \gamma(z), }[/math]

or, for some countable orthonormal basis (ei)i∈N of H,

[math]\displaystyle{ Sx = \sum_{i \in \mathbf{N}} \sigma_{i}^{2} \langle x, e_{i} \rangle e_{i}. }[/math]

In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0 < q < 1 such that

[math]\displaystyle{ \sigma_{i + 1}^{2} \leq q \sigma_{i}^{2}, }[/math]

then, for all f ∈ L1(H, γ; R),

[math]\displaystyle{ \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu(y) \xrightarrow[r \to 0]{\gamma} f(x), }[/math]

where the convergence is convergence in measure with respect to γ. In 1988, Tišer showed that if

[math]\displaystyle{ \sigma_{i + 1}^{2} \leq \frac{\sigma_{i}^{2}}{i^{\alpha}} }[/math]

for some α > 5 ⁄ 2, then

[math]\displaystyle{ \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu(y) \xrightarrow[r \to 0]{} f(x), }[/math]

for γ-almost all x and all f ∈ Lp(H, γ; R), p > 1\.

As of 2007, it is still an open question whether there exists an infinite-dimensional Gaussian measure γ on a separable Hilbert space H so that, for all f ∈ L1(H, γ; R),

[math]\displaystyle{ \lim_{r \to 0} \frac1{\gamma \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \gamma(y) = f(x) }[/math]

for γ-almost all x ∈ H. However, it is conjectured that no such measure exists, since the σi would have to decay very rapidly.

## See also

* Differentiation rules - Rules for computing derivatives of functions
* Leibniz integral rule - Differentiation under the integral sign formula
* Reynolds transport theorem - 3D generalization of the Leibniz integral rule

## References

* Preiss, David; Tišer, Jaroslav (1982). "Differentiation of measures on Hilbert spaces". Measure theory, Oberwolfach 1981 (Oberwolfach, 1981). Lecture Notes in Mathematics. 945. Berlin: Springer. pp. 194–207. doi:10.1007/BFb0096675. ISBN 978-3-540-11580-9.
* Tišer, Jaroslav (1988). "Differentiation theorem for Gaussian measures on Hilbert space". Transactions of the American Mathematical Society 308 (2): 655–666. doi:10.2307/2001096. https://www.ams.org/journals/tran/1988-308-02/S0002-9947-1988-0951621-8/S0002-9947-1988-0951621-8.pdf.

* v
* t
* e

Calculus

Precalculus|

* Binomial theorem
* Concave function
* Continuous function
* Factorial
* Finite difference
* Free variables and bound variables
* Fundamental theorem
* Graph of a function
* Linear function
* Mean value theorem
* Radian
* Rolle's theorem
* Secant
* Slope
* Tangent

Limits|

* Indeterminate form
* Limit of a function
* One-sided limit
* Limit of a sequence
* Orders of approximation
* (ε, δ)-definition of limit

Differential calculus|

* Chain rule
* Derivative
* Differential
* Differential equation
* Differential operator
* Implicit differentiation
* Inverse functions and differentiation
* L'Hôpital's rule
* Leibniz's rule
* Logarithmic derivative
* Mean value theorem
* Newton's method
* Notation
* Leibniz's notation
* Newton's notation
* Product rule
* Quotient rule
* Regiomontanus' angle maximization problem
* Related rates
* Simplest rules
* Constant factor rule in differentiation
* Linearity of differentiation
* Power rule
* Sum rule in differentiation
* Stationary point
* First derivative test
* Second derivative test
* Extreme value theorem
* Maxima and minima
* Taylor's theorem

Integral calculus|

* Antiderivative
* Arc length
* Constant of integration
* Differentiation under the integral sign
* Fundamental theorem of calculus
* Integral of secant cubed
* Integral of the secant function
* Integration by parts
* Integration by substitution
* Tangent half-angle substitution
* Partial fractions in integration
* Quadratic integral
* Proof that 22/7 exceeds π
* Simplest rules
* Constant factor rule in integration
* Linearity of integration
* Sum rule in integration
* Trapezoidal rule
* Trigonometric substitution

Vector calculus|

* Curl
* Directional derivative
* Divergence
* Divergence theorem
* Gradient
* Gradient theorem
* Green's theorem
* Laplacian
* Stokes' theorem

Multivariable calculus|

* Curvature
* Disc integration
* Divergence theorem
* Exterior
* Gabriel's Horn
* Geometric
* Hessian matrix
* Jacobian matrix and determinant
* Line integral
* Matrix
* Multiple integral
* Partial derivative
* Shell integration
* Surface integral
* Tensor
* Volume integral

Series|

* Abel's test
* Alternating
* Alternating series test
* Arithmetico-geometric sequence
* Binomial
* Cauchy condensation test
* Direct comparison test
* Dirichlet's test
* Euler–Maclaurin formula
* Fourier
* Geometric
* Harmonic
* Infinite
* Integral test for convergence
* Limit comparison test
* Maclaurin
* Power
* Ratio test
* Root test
* Taylor
* Term test

Special functions and numbers|

* Bernoulli numbers
* e (mathematical constant)
* Exponential function
* Natural logarithm
* Stirling's approximation

History of calculus|

* Adequality
* Brook Taylor
* Colin Maclaurin
* Generality of algebra
* Gottfried Wilhelm Leibniz
* Infinitesimal
* Infinitesimal calculus
* Isaac Newton
* Fluxion
* Law of Continuity
* Leonhard Euler
* Method of Fluxions
* The Method of Mechanical Theorems

Lists|

* Differentiation rules
* List of integrals of exponential functions
* List of integrals of hyperbolic functions
* List of integrals of inverse hyperbolic functions
* List of integrals of inverse trigonometric functions
* List of integrals of irrational functions
* List of integrals of logarithmic functions
* List of integrals of rational functions
* List of integrals of trigonometric functions
* List of limits
* List of mathematical symbols
* Lists of integrals

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Differentiation of integrals. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Differentiation_of_integrals&oldid=3363856"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template