Lebesgue differentiation theorem

In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis.

tatement

For a Lebesgue integrable real-valued function $f$, the indefinite integral is a set function which maps a measurable set $A$ to the Lebesgue integral of $f\; cdot\; mathbf\{1\}\_A$, where $mathbf\{1\}\_\{A\}$ denotes the characteristic function of the set $A$. It is usually written

::$int\_\{A\}f\; dlambda.$

We define the "derivative" of this integral at $x$ to be

::$lim\_\{B\; ightarrow\; x\}\; frac\{1\}\; int\_\{B\}f\; dlambda,$

where "B" is a ball centered at $x$, and $B\; ightarrow\; x$ means that the diameter of $B$ tends to zero. The Lebesgue differentiation theorem states that this derivative is equal to $f$ almost everywhere. Since functions which are equal almost everywhere have the same integral over any set, this result is the best possible in the sense of recovering the function from integrals.

A more general version also holds. We may replace the balls $B$ by sets $U$ of bounded eccentricity. This means we simply require that each set $U$ is contained in a ball $B$ centered at $x$, with $lambda(U)\; geq\; clambda(B)$, for some fixed $c\; >\; 0$. If we let these sets shrink to $x$, the same result holds.

Proof

The usual proof of this theorem requires the Vitali covering lemma (e.g. see "Measure and Integral"). The theorem also holds if cubes are replaced by arbitrary sets with diameter tending to zero, in the definition of the derivative. This follows since the same substitution can be made in the statement of the Vitali covering lemma.

Discussion

This is an analogue, and a generalization, of the fundamental theorem of calculus, which equates a Riemann integrable function and the derivative of its (indefinite) integral. It is also possible to show a converse - that every differentiable function is equal to the integral of its derivative, but this requires a Henstock-Kurzweil integral in order to be able to integrate an arbitrary derivative.

A special case of the Lebesgue differentiation theorem is the Lebesgue density theorem, which is equivalent to the differentiation theorem for characteristic functions of measurable sets. The density theorem is usually proved using a simpler method (eg see Measure and Category).

A more general version also holds, where we replace the cubes $Q$ by more general sets

ee also

* Differentiation of integrals

References

* Measure and Integral - An introduction to Real Analysis, Richard L. Wheeder & Antoni Zygmund, Dekker, 1977
* Measure and Category, John C. Oxtoby, Springer-Verlag, 1980
* Princeton Lectures in Analysis III: Real Analysis, Elias M. Stein & Rami Shakarchi, Princeton University Press, 2005