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exists. In that case, we call the strong derivative, classical derivative, or just derivative for short, of at . We say that is everywhere differentiable, or differentiable for short, if it is differentiable at all points , and differentiable almost everywhere if it is differentiable at almost every point . If is differentiable everywhere and its derivative is continuous, then we say that is continuously differentiable.
Remark 1 Much later in this sequence, when we cover the theory of distributions, we will see the notion of a weak derivative or distributional derivative, which can be applied to a much rougher class of functions and is in many ways more suitable than the classical derivative for doing “Lebesgue” type analysis (i.e. analysis centred around the Lebesgue integral, and in particular allowing functions to be uncontrolled, infinite, or even undefined on sets of measure zero). However, for now we will stick with the classical approach to differentiation.
Exercise 2 If is everywhere differentiable, show that is continuous and is measurable. If is almost everywhere differentiable, show that the (almost everywhere defined) function is measurable (i.e. it is equal to an everywhere defined measurable function on outside of a null set), but give an example to demonstrate that need not be continuous.
Exercise 3 Give an example of a function which is everywhere differentiable, but not continuously differentiable. (Hint: choose an that vanishes quickly at some point, say at the origin , but which also oscillates rapidly near that point.)
In single-variable calculus, the operations of integration and differentiation are connected by a number of basic theorems, starting with Rolle’s theorem.
Theorem 4 (Rolle’s theorem) Let be a compact interval of positive length, and let be a differentiable function such that . Then there exists such that .
Proof: By subtracting a constant from (which does not affect differentiability or the derivative) we may assume that . If is identically zero then the claim is trivial, so assume that is non-zero somewhere. By replacing with if necessary, we may assume that is positive somewhere, thus . On the other hand, as is continuous and is compact, must attain its maximum somewhere, thus there exists such that for all . Then must be positive and so cannot equal either or , and thus must lie in the interior. From the right limit of (1) we see that , while from the left limit we have . Thus and the claim follows.
Remark 5 Observe that the same proof also works if is only differentiable in the interior of the interval , so long as it is continuous all the way up to the boundary of .
Exercise 6 Give an example to show that Rolle’s theorem can fail if is merely assumed to be almost everywhere differentiable, even if one adds the additional hypothesis that is continuous. This example illustrates that everywhere differentiability is a significantly stronger property than almost everywhere differentiability. We will see further evidence of this fact later in these notes; there are many theorems that assert in their conclusion that a function is almost everywhere differentiable, but few that manage to conclude everywhere differentiability.
Remark 7 It is important to note that Rolle’s theorem only works in the real scalar case when is real-valued, as it relies heavily on the least upper bound property for the domain . If, for instance, we consider complex-valued scalar functions , then the theorem can fail; for instance, the function defined by vanishes at both endpoints and is differentiable, but its derivative is never zero. (Rolle’s theorem does imply that the real and imaginary parts of the derivative both vanish somewhere, but the problem is that they don’t simultaneously vanish at the same point.) Similar remarks to functions taking values in a finite-dimensional vector space, such as .
One can easily amplify Rolle’s theorem to the mean value theorem:
Corollary 8 (Mean value theorem) Let be a compact interval of positive length, and let be a differentiable function. Then there exists such that .
Proof: Apply Rolle’s theorem to the function .
Remark 9 As Rolle’s theorem is only applicable to real scalar-valued functions, the more general mean value theorem is also only applicable to such functions.
Exercise 10 (Uniqueness of antiderivatives up to constants) Let be a compact interval of positive length, and let and be differentiable functions. Show that for every if and only if for some constant and all .
We can use the mean value theorem to deduce one of the fundamental theorems of calculus:
Theorem 11 (Second fundamental theorem of calculus) Let be a differentiable function, such that is Riemann integrable. Then the Riemann integral of is equal to . In particular, we have whenever is continuously differentiable.
Proof: Let . By the definition of Riemann integrability, there exists a finite partition such that
for every choice of .
Fix this partition. From the mean value theorem, for each one can find such that
and thus by telescoping series
Since was arbitrary, the claim follows.
Remark 12 Even though the mean value theorem only holds for real scalar functions, the fundamental theorem of calculus holds for complex or vector-valued functions, as one can simply apply that theorem to each component of that function separately.
Of course, we also have the other half of the fundamental theorem of calculus:
Theorem 13 (First fundamental theorem of calculus) Let be a compact interval of positive length. Let be a continuous function, and let be the indefinite integral . Then is differentiable on , with derivative for all . In particular, is continuously differentiable.
Proof: It suffices to show that
for all , and
for all . After a change of variables, we can write
for any and any sufficiently small , or any and any sufficiently small . As is continuous, the function converges uniformly to on as (keeping fixed). As the interval is bounded, thus converges to , and the claim follows.
for all ,
for all , and thus
for all .
In these notes we explore the question of the extent to which these theorems continue to hold when the differentiability or integrability conditions on the various functions are relaxed. Among the results proven in these notes are
- The Lebesgue differentiation theorem, which roughly speaking asserts that Corollary 14 continues to hold for almost every if is merely absolutely integrable, rather than continuous;
- A number of differentiation theorems, which assert for instance that monotone, Lipschitz, or bounded variation functions in one dimension are almost everywhere differentiable; and
- The second fundamental theorem of calculus for absolutely continuous functions.
The material here is loosely based on Chapter 3 of Stein-Shakarchi. Read the rest of this entry »
For these notes, is a fixed measurable space. We shall often omit the -algebra , and simply refer to elements of as measurable sets. Unless otherwise indicated, all subsets of X appearing below are restricted to be measurable, and all functions on X appearing below are also restricted to be measurable.
We let denote the space of measures on X, i.e. functions which are countably additive and send to 0. For reasons that will be clearer later, we shall refer to such measures as unsigned measures. In this section we investigate the structure of this space, together with the closely related spaces of signed measures and finite measures.
Suppose that we have already constructed one unsigned measure on X (e.g. think of X as the real line with the Borel -algebra, and let m be Lebesgue measure). Then we can obtain many further unsigned measures on X by multiplying m by a function , to obtain a new unsigned measure , defined by the formula
If is an indicator function, we write for , and refer to this measure as the restriction of m to A.
Exercise 1. Show (using the monotone convergence theorem) that is indeed a unsigned measure, and for any , we have . We will express this relationship symbolically as
Exercise 2. Let m be -finite. Given two functions , show that if and only if for m-almost every x. (Hint: as usual, first do the case when m is finite. The key point is that if f and g are not equal m-almost everywhere, then either f>g on a set of positive measure, or f<g on a set of positive measure.) Give an example to show that this uniqueness statement can fail if m is not -finite. (Hint: take a very simple example, e.g. let X consist of just one point.)
In view of Exercises 1 and 2, let us temporarily call a measure differentiable with respect to m if (i.e. ) for some , and call f the Radon-Nikodym derivative of with respect to m, writing
by Exercise 2, we see if is -finite that this derivative is defined up to m-almost everywhere equivalence.
Exercise 3. (Relationship between Radon-Nikodym derivative and classical derivative) Let m be Lebesgue measure on , and let be an unsigned measure that is differentiable with respect to m. If has a continuous Radon-Nikodym derivative , show that the function is differentiable, and for all x.
Exercise 4. Let X be at most countable. Show that every measure on X is differentiable with respect to counting measure .
If every measure was differentiable with respect to m (as is the case in Exercise 4), then we would have completely described the space of measures of X in terms of the non-negative functions of X (modulo m-almost everywhere equivalence). Unfortunately, not every measure is differentiable with respect to every other: for instance, if x is a point in X, then the only measures that are differentiable with respect to the Dirac measure are the scalar multiples of that measure. We will explore the precise obstruction that prevents all measures from being differentiable, culminating in the Radon-Nikodym-Lebesgue theorem that gives a satisfactory understanding of the situation in the -finite case (which is the case of interest for most applications).
In order to establish this theorem, it will be important to first study some other basic operations on measures, notably the ability to subtract one measure from another. This will necessitate the study of signed measures, to which we now turn.
[The material here is largely based on Folland’s text, except for the last section.]