A Computational Aspect of the Lebesgue Differentiation Theorem

Given an L 1-computable function, f , we identify a canonical representative of the equivalence class of f , where f and g are equivalent if and only if |f − g| is zero. Using this representative, we prove a modified version of the Lebesgue Differentiation Theorem. Our theorem is stated in terms of Martin-Löf random points in Euclidean space.


Introduction
The Lebesgue Differentiation Theorem is a fundamental theorem in measure theory which generalizes the fundamental theorem of calculus.The Lebesgue Differentiation Theorem states that given f ∈ L 1 ([0, 1] d ), for almost every x where Q is a cube in [0, 1] d containing x.In this paper, we look at the theorem in the context of computability theory.A proof of the Lebesgue Differentiation theorem can be found in the book of Wheeden-Zygmund [6] (p. 101-109) and with some work we can modify this proof for L 1 -computable functions which are defined in Theorem 2.1.The final result we obtain will be a modified version of the Lebesgue Differentiation Theorem and will hold for all x which are Martin-Löf random.Due to the nature of Lebesgue integration, rather than working with actual functions f , it will be more useful to work with canonical representatives of f based on the equivalence relation In this paper, we will prove such a canonical representative exists, and is well defined.Eventually, upon using some ideas from the original proof and creating some new

Some Notation and Definitions
The functions we consider in this paper will be real-valued and Lebesgue measurable with measure µ on the unit cube [0, 1] d ⊂ R d .Here d is a fixed positive integer.Our short description of this will be f ∈ L 1 ([0, 1] d ).Our norm will be the L 1 -norm which is defined by Note that L 1 functions have finite Lebesgue integral.
The following definition was provided by Pour-El and Richards [4].
For future reference, note that we can easily find a computable sequence of rational numbers (D n ), depending on f n , where D n is an upper bound of the maximum gradient of each x with edges parallel to the coordinate axes.Then, the indefinite integral of f is said to be differentiable at x if Lemma 2.3 Given a rational cube Q ⊆ [0, 1] d and an L 1 -computable function f , we can effectively find the computable real number Q f .Proof To show that Q f exists and is computable we need to find a recursive sequence of rational numbers, (c n ) such that for all n.By Theorem 2.1, there exists a computable sequence of polynomials with rational coefficients, f n , such that We want to say that c n = Q f n .First we show that Q f n is rational for all n.To do this, recall that a polynomial is just a sum of monomials and since the integral of a sum is just the sum of integrals, we can just consider the integral of a monomial over Q.Note that in the following calculation p l ≤ q l for all l and (q 1 , . . ., q d ) and (p 1 , . . ., This is a rational number because by the definition of f n , a must be rational and q 1 , p 1 , . . ., q d , p d are also rational because they are the coordinates of our rational cube Q.Thus, we see that Q f n is rational for all n.Also, since (f n ) is a recursive sequence, ( Q f n ) is also a recursive sequence.We say this because Q f n can be re-written as the integral over Q of the elements of the sequence (f n ).Now that we have our recursive sequence of rational numbers, we just need to show that This proves that Q f exists and is computable.
The next proposition provides a useful property of Σ 0 1 sets.

Proposition 2.6
The class of Σ 0 1 sets is closed under the existential number quantifier.
The next definition was first given by Martin-Löf [3].
Another characterization of random points in R d is given by Solovay's Lemma, as proven by Simpson [5].The proof given by Simpson [5] is given for sets in the Cantor space, but the proof applies here as well.
Then for any random x ∈ [0, 1] d , x lies in only finitely many V k .
Before we begin the proof of our modified Lebesgue Differentiation Theorem, we will need a few concepts and results to help set up the proof of the theorem. Then The result follows.
The next lemma is based on Proposition 4.1 found in a paper by Brown, Guisto and Simpson [1].
and for all x / ∈ V k and n ≥ k we have for all i ≥ 2n.
Proof Let f n and D n be as in Theorem 2.1.
We want to show that the V k are Σ 0 1 .Since f n is continuous for all n, x ∈ V k if and only if there exists a ball around x contained in V k .For this reason, we can rewrite V k as follows, Here n and i are natural numbers and a = a is a recursive predicate, so by Theorem 2.4 and Theorem 2.5 and Theorem 2.6, we can see that Now, we need to look at the measure of V k .Note that Using the previous lemma, we can conclude the proof as follows: Thus we have our V k and by the definition of V k , for all x not in V k and n ≥ k, for all i ≥ n Lemma 3.3 Let f be L 1 -computable.Then lim n→∞ f n (x) exists for all random x.
Proof From the previous lemma, we can see that Since the sum is finite, we can use Solovay's Lemma and say that any random x will only be in finitely many V k .So, for some large k and ∀n ≥ k, we can see that for all i ≥ 2n.From this we can see that f n converges uniformly for x / ∈ V k , and the limit exists.
By Theorem 3.3 lim n→∞ f n (x) exists for all random x so we know our new function is well-defined.We want to claim that f is a canonical representation of the equivalence class of f (f ∼ g ⇔ ||f − g|| 1 = 0).The next two lemmas will prove that this is indeed the case.
> ε and consider the set E 0 .If this is a set of measure zero, then the result follows.Suppose, however, that this is not the case.Then, there exists some small ε > 0 such that µ{E ε } > ε.Now, by Theorem 3.2, for all random x, there exists k large such that x / ∈ V k and for all n ≥ k, for all i ≥ k.By the definition of f we can also say that f (x) − f 2n (x) ≥ 1 2 n .So for n such as the one above and x / ∈ V k , This is a contradiction because ||f − f 2n || 1 → 0 at n → ∞.It follows that our assumption is incorrect and therefore, E 0 has measure zero.
A computational aspect of the Lebesgue differentiation theorem 9 Proof (⇐) Suppose f = g.Then, f (x) = g(x) for all x except for a set of measure zero.The result follows.
n This is a useful fact that will be used a little later.First we look at another consequence of our given assumption.
We would like to show that there cannot be any random x ∈ E. Let The sets V k n will be our test for randomness.Using sets similar to the ones used in the proof of Theorem 3.2, we can show that for a fixed k the sequence (V k n ) is uniformlyΣ 0 1 .We would now like to use Solovay's Lemma.To do that, we need to show that the sum of the measures of V k n over all n is finite.To do this, we will use Theorem 3.1 again.
For a fixed k this is a geometric series, so ∞ n=1 V k n < ∞.By Solovay's Lemma, for any fixed k, x can only be in finitely many V k n .Therefore for a random x, 2 k for all k meaning that lim n→∞ |f n (x) − g n (x)| = 0.This shows that there cannot be a random x in E. By the definition of f and g, there cannot be a non-random x in E either.This means that E is empty, and f = g From the last two lemmas, we can see that f is a canonical representative of the equivalence class of f .The next section provides some results necessary for the proof of the main theorem.

Some Important Lemmas
First, will will prove the Lebesgue Differentiation theorem for continuous functions.Proposition 4.1 Let f ∈ L 1 R d be a continuous function.Then, the indefinite integral of f is differentiable and its derivative is equal to f (x) for all x ∈ R d .
Proof The proof is clear from the following calculations.If f is continuous at x and Q is a rational cube containing x, then which tends to zero as Q shrinks to x.
Using this fact we develop the idea for the proof.By Theorem 2.1, we can approximate our function f using continuous polynomials.Using this, we can approximate the indefinite integral of f and create a test for randomness.To do all this, we will need a few lemmas.
A proof of the Simple Vitali Lemma is given by Wheedon and Zygmund [6] on page 102.
where the supremum is taken over every cube Q with center x and edges parallel to the coordinate axes.This function, g * is called the Hardy-Littlewood maximal function of g.
This is a slightly modified version of the definition of the Hardy-Littlewood maximal function.
Moreover, there is a constant c independent of f and α such that Proof Since the domain of f and f * is [0, 1] d if we fix α > 0 we can say that the measure of the set The collection of such Q x covers E, so by Theorem 4.2, there exist β > 0 and x 1 , . . .
Putting everything together, we get This proves the Hardy-Littlewood Lemma.
Definition 4.6 Given f ∈ L 1 and ε > 0, let S * (f , ε) be the union of all Q such that Note that according to this definition, and the Hardy-Littlewood Lemma, Lemma 4.7 Let c be the constant from the Hardy-Littlewood Lemma and let f be L 1 -computable.Then, we can find sets V * k which are uniformly Since the union of countable many uniformly Σ 0 1 sets is Σ 0 1 , we need to show that the sequence Then, we can write S * f − f 2n , 1 2 n in the following way: This is a sequence of uniformly Σ 0 1 sets by Theorem 2.5 which means that each V * k is Σ 0 1 and the sequence (V * k ) is uniformly Σ 0 1 .Now we need to look at the measure of V * k .We will use the Hardy-Littlewood Lemma.
The result follows.
There is one last lemma we need before we can prove the main result.
for all rational cubes Q containing x.Here the sequence of V k is from Theorem 3.2 and the sequence of V * k is from Theorem 4.7.
Proof Since D n is an upper bound of the maximum gradient of each f n , max{|∇f n | : x ∈ [0, 1] d }, we can use the Mean Value Theorem to say, for all rational Q containing x.By Theorem 3.2 we have that 2 n and by Theorem 4.7, Noopur Pathak for all n ≥ k and x / ∈ V k ∪ V * k .Combining these two, we get that Proof Let f n and D n be as in Theorem 2.1.V k from Theorem 3.2 and V * k from Theorem 4.7 form Martin-Löf tests.So, for a random x, there exists a large k such that x / ∈ V k ∪ V * k .We want to show that for all ε > 0 ∃ δ > 0 such that whenever the diameter of Q is less than δ .Choose n large so that 1 2 n−1 < ε 2 and let δ = ε 2D 2n .Then, when the diameter of Q is less than δ , By only dealing with L 1 -computable functions, our theorem seems at first to be less general than the original Lebesgue Differentiation Theorem.However, if we consider relativization, it can be seen that the statement proved in this paper is stronger.Any L 1 function is computable relative to some oracle and using this we can prove a relativized version of Theorem 5.1 pertaining to any function and provide a very specific set of measure zero, outside of which the Lebesgue Differentiation Theorem always holds.
In Theorem 5.1 we have proved that the Lebesgue Differentiation Theorem holds at x provided that x is a Martin-Löf random point in a Euclidean space.The natural question arises, is the converse true?That is, if we have that the Lebesgue Differentiation Theorem holds at x for all L 1 -computable functions, is x necessarily random?This is an important question as the converse holding would give an alternative characterization of random points in Euclidean space.

Lemma 4 . 2 (
Simple Vitali Lemma) Let E be a subset of [0, 1] d , and let K be a collection of cubes Q in [0, 1] d covering E. Then there exists a positive constant β , depending only on d , and a finite number of disjoint cubes Q 1

Lemma 4 . 8
Let f be L 1 -computable and let D n and f n be as in Theorem 2.1.Then for all k, n ≥ k and all x /

Theorem 5 . 1
Let f be an L 1 -computable function.Let f be the canonical representation of f as defined in Theorem 3.4.Then for all random x, f