On the $L^1$ and pointwise divergence of continuous functions

For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$ \lim_{n \to \infty} \int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means.


Introduction
Quasiarihmetic means were introduced in 1920-s/30-s by de Finetti [7], Knopp [12], Kolmogorov [13] and Nagumo [16].For a continuous and strictly monotone function F : I → R (here and below I stands for an arbitrary subinterval of R and CM(I) stands for a family of all continuous and strictly monotone functions on I) we define the quasiarithmetic mean A [F ] : ∞ n=1 I n → I by where n ∈ N and a = (a 1 , . . ., a n ) ∈ I n .
A function F is called a generating function or a generator of A [F ] .It was Knopp [12] who noticed that for I = R + , π p (x) := x p (p = 0) and π 0 (x) := ln x, the quasiarithmetic mean A [πp] coincides with the p-th power mean P p .
Adapting the classical result lim p→+∞ P p = max we say that a family (F n ) ∞ n=1 in CM(I) is QA-maximal provided lim n→∞ A [Fn] = max pointwise.
There are few approaches to this property.First, applying some general results by Páles [27], we can establish the general equivalent condition of being QA-maximal (see also [19]).More precisely, a sequence (F n ) ∞ n=1 of elements in CM(I) is a QAmaximal family if and only if = 0 for all x, y, z ∈ I with x < y < z.
It is a particular case of an analogous result for deviation and quasideviation means -see the papers by Daróczy [2,3], Daróczy-Losonczi [4], Daróczy-Páles [5,6], and by Páles [20][21][22][23][24][25][26] for detailed study of these families.It turns out that under the additional assumption that each generator is twice continuously differentiable with nowhere vanishing first derivative -from now on we denote family of all such generators by C 2# (I) -we can establish another equivalent conditions.More precisely, by [19], a family (F n ) ∞ n=1 of elements in C 2# (I) such that is uniformly lower bounded is QA-maximal if and only if dt = +∞ for all x, y ∈ I with x < y.
Let us emphasize that the operator F ′′ F ′ plays a key role in a comparability of quasiarithmetic means.More precisely, by Jensen inequality, for all F, G ∈ C 2# (I) we have [17], the maximal property is connected with the set Namely, it was proved that if Conversely, for every QA-maximal family (F n ) ∞ n=1 the set I F is a dense subset of I.These results were strengthened in [19].More precisely, there is proved that if the intersection of I F with an arbitrary open subset of I has a positive Lebesgue measure λ, then (F n ) ∞ n=1 is QA-maximal.This assumption is somehow the weakest possible, as for every X ⊂ I such that λ(X ∩ J) = 0 for some open subinterval J of I there exists a family (G n ) ∞ n=1 which is not QA-maximal, however X ⊂ I G .On the other hand, it was proved that I Z could have the Hausdorff dimension zero for some QA-maximal family In what follows our aim is to study the relation between the property of being a max-family and the corresponding set I.
Let us emphasize that the same consideration remains valid for QA-minimal and min-families (with a natural definition).As a matter of fact we can reapply all results below to the reflected means -for detailed study of reflected means we refer the reader to recent development by Chudziak-Páles-Pasteczka [1], Páles-Pasteczka [29] and Pasteczka [18].
1.1.Rephrasing of the problem.As all conditions above are expressed in terms of the operator F → F ′′ /F ′ we are going to elaborate the properties of this operator.To this end, for a sequence of continuous functions f 1 , f 2 , . . .: Furthermore, let Ω(I) be a family of all possible I f -s.More precisely, f n (t)dt = +∞ for all x, y ∈ I with x < y, then the family (f n ) ∞ n=1 is called max-family.Based on the previous section we obtain the following results.
Proposition 1 ( [19], Proposition 4.1).Let X ⊂ I be an arbitrary set.If λ(X ∩J) = 0 for some open interval J, then there exists a sequence (f n : n=1 be a family of continuous functions with f 1 ≤ f 2 ≤ . . .and such that I f intersected with each open subinterval of I has a positive Lebesgue measure.Then (f n ) is a max-family.

Propositions above motivates us to define
n=1 is a max-family .Obviously Ω 0 (I) ⊆ Ω(I).The aim of this paper is to show several important facts concerning the set Ω(I), Ω 0 (I), and their common relations.

General properties of Ω(I) and Ω 0 (I)
First, let us present our initial result which shows that Ω(I) consists of G δ sets only.
n=1 be a family of continuous function with Let us now prove that the set Ω(I) is closed under finite union and closed under countable intersection (with additional assumption).

Lemma 2. Let I be an interval and (f
To prove the converse inclusion take x ∈ I f +g arbitrarily.Then we have which shows that lim n→∞ f n (x) = +∞ or lim n→∞ g n (x) = +∞, i.e. x ∈ I f ∪ I g .
Observe that Lemma 2 fails to be true for countable sequence of families of continuous functions.To see that set take two sequences of families f n ∞ n=1 , . . . on R defined by f (1)  n ≡ n, for all n ∈ N; for all n ∈ N and i ≥ 2, Then we have On the other hand Above equalities show that both inclusions may fail.
Let us note as a curiosity the following remark that mimics Lemma 2.
Remark 1.Let I be an interval and (f n : Proof.Take x ∈ I f •g arbitrarily.Then we have which is possible when lim n→∞ f n (x) and lim n→∞ g n (x) are non-negative and at least one of them is +∞.Hence x ∈ I f ∪ I g .
To show the converse inclusion assume that x ∈ I f and lim Lemma 4 ( [19], Proposition 4.3).For every interval I there exists a max-family (z n : I → R + ) ∞ n=1 such that dim H I z = 0.In particular Ω 0 (I) contains a set of zero Hausdorff dimension.

Max-families with noninteger Hausdorff dimension
In the theory of fractals two natural questions are present: Is there a set which Hausdorff dimension equals the given number?And if the answer is positive: What additional features this set can have?The answers to the first question was given for instance by [10,30,31], while the answer to the second question depends on the feature (see [32] for connections with ergodicity and continued fractions, [9] for properties of distance sets and [14] for examples of subrings of R).
In this section we present two construction of max-families with an arbitrary Hausdorff dimension θ ∈ (0, 1).In the first (Cantor-type) approach we show that Ω 0 (I) contains a set which can be factorized to a nowhere dense set of Hausdorff dimension θ and a dense set of Hausdorff dimension zero.In the second (Jarník-type) approach we construct a set in I f ∈ Ω 0 (I) such that dim H (I f ∩ U) = θ for an arbitrary open interval U ⊂ I.We provide two constructions; the Cantor-like sets can be described directly, while the second one relies on number-theoretical approach and thus does not give any insight on how does the provided set actually look like.

3.1.
Cantor-type construction.We recall some basic notation and definitions from the fractal theory.We call a function f : X → X a contraction if it is Lipschitz with constant c f ∈ (−1, 1).We call a finite set F = {f 1 , . . ., f n } of contractions defined on a compact metric space X an iterated function system, or IFS.We say that the IFS F satisfies the open set condition (abbreviated OSC ) if there exists an open and bounded set V = ∅ with F (V ) ⊂ V , where F (V ) := f 1 (V ) ∪ . . .∪ f n (V ) and f i (V ) ∩ f j (V ) = ∅ for all distinct i, j ∈ {1, . . .n}. Proposition 3 (Moran [15]).Suppose that F satisfies the open set condition and each f i ∈ F is a similarity with contraction constant c i .If A is the compact set such that F (A) = A, then dim H A = s, where s is the unique solution of the equation Lemma 5.For every interval I and every θ ∈ (0, 1) there exists a family Proof.One can assume without loss of generality that I = [0, 1].Let m := ( 12 ) 1/θ ∈ (0, 1  2 ).Take ε ∈ (0, 1 2m − 1) and define an IFS Then F L and F R are similarities with Lipschitz constants m, F Obviously each D n is a closed subset of I. Now we prove that the sequence satisfies all conditions (i)-(iv).Condition (i) is obvious.To show the second property observe that Moreover if D n+1 ⊂ int D n for some n ∈ N, then as both F L and F R are homeomorphisms we get and thus D n+2 ⊂ int D n+1 .By simple induction we obtain property (ii).Denote D ∞ = ∞ n=0 D n .Condition (iii) follows from (ii) and the general theory of fractal sets (see for instance Theorem 9.1 in [8]).
To check the last condition denote s := dim H D ∞ .Then, by Proposition 3 we have 2m s = 1.Thus s = ln(1/2) ln m = θ, which is (iv).Remark 2. In Lemma 5 the set D ∞ has Lebesgue measure zero.It turns out that the entire construction presented there is equivalent to the construction of the uniform Cantor set on the interval 1−m removed (that is both constructions lead to the same set).Proof.For θ = 0 the statement is an easy implication of Lemma 4. Similarly, for θ = 1 we can take d n ≡ n.
For θ ∈ (0, 1) set a family (D n ) ∞ n=0 like in Lemma 5.By Lemma 3 we get that which is easily equivalent to our statement.Theorem 1.For every θ ∈ [0, 1], there exists a max-family (f n : I → R) ∞ n=1 such that dim H I f = θ and I f can be decomposed to a nowhere dense set and a set of Hausdorff dimension zero.Therefore it is sufficient to show that (f n ) ∞ n=1 is a max-family.As (z n ) ∞ n=1 is a max-family we have 0 ≤ z 1 ≤ z 2 ≤ . . . .Binding this property with (3.1) and the definition of the sequence (f n ) ∞ n=1 we easily obtain Thus the only remaining part to be proved is that (1.1) holds.However, as d n ≥ 0 for all n ∈ N and (z n ) ∞ n=1 is a max-family, we obtain

Jarník-type construction.
In what follows we show that for every θ ∈ (0, 1) there exists a max-family (f n : I → R) ∞ n=1 such that dim H (I f ∩ J) = θ for every open subinterval J ⊂ I.The important theorem concerning neighbourhoods of rational numbers will be used.
Then, as • is continuous, we have cl Y q,α ⊂ int Z q,α for all q ∈ N and α > α 0 .Now let δ q : I → [0, 1] be defined by Then we obtain I r ⊇ {x ∈ I : x ∈ Y q,α 0 for infinitely many q ∈ N} = Q α 0 .
On the other hand for all α < α 0 there exists a number q 0 ∈ N such that q+1 q q 1−α 0 ≤ q 1−α for all q ≥ q 0 .
Finally we have Therefore by Proposition 4 we obtain that for every open interval U ⊂ I we have Consequently, dim H (I r ∩ U) = 2 α 0 = θ for every open subinterval U ⊂ I. Now let (z n ) ∞ n=1 be a family from Lemma 4 and let f n := z n + r n for n ∈ N.Then, as r n ≥ 0, we obtain that (f n ) ∞ n=1 is a max-family.Furthermore by Lemma 2 we get that I f ⊃ I r and I f \I r is of Hausdorff dimension zero which completes the proof.
Final conclusions and remarks.At the very end let us put the reader's attention to few important problems.First, we cannot exclude that Ω(I) is a family of all G δ subsets of I and/or Ω 0 (I) contains all dense G δ subsets of I.In particular, it is interesting to find a full characterization of all elements of sets Ω(I) and Ω 0 (I) (our results show that they can be complicated from the measure-theoretical point of view).Second, this problem has a natural multidimensional generalization where the domain of the integral in (1.1) is taken over all open subsets of a given domain.Finally, it is not known if the assumption f 1 ≤ f 2 ≤ . . . in the definition of Ω(I) can be relaxed.

Lemma 3 .
g and similarly, I g ⊆ I f •g .Let I be an interval and (D n ) ∞ n=0 be a family of closed subsets of I such that D 0 = I and D n+1 ⊂ int D n for all n ∈ N + ∪ {0}.Then ∞ n=0 D n ∈ Ω(I).Proof.Indeed, in view of Tietze(-Urysohn-Brouwer) theorem, for every n ∈ N there exists a continuous function δ n : I → [0, 1] such that δ n (x) = 0 for x ∈ I \ D n ; 1 for x ∈ D n+1 .Define d n := n i=0 δ n .Then for every x ∈ ∞ k=0 D k =: D ∞ we have d n (x) = n.In particular I d ⊇ D ∞ .Now fix N ∈ N. Then as D n ⊆ D N for n ≥ N, we have δ n (x) = 0 for x ∈ I \ D N .Thus d n (x) < N for all n ≥ N and x ∈ I \ D N .This proves that I d ⊆ D N .As N was an arbitrary natural number we get I d ⊆ D ∞ , which completes the equality and thus F satisfies the OSC with U = int I. Now define a family (D n ) ∞ n=0 by D 0 := I D n+1 := F (D n ) for n ≥ 0.

Proof.
Take θ ∈ [0, 1] arbitrarily.By Lemma 4 one can take a max-family (z n : I → R + ) ∞ n=1 such that dim H I z = 0. Furthermore, by Lemma 6 we can take a family (d n :I → R + ) ∞ n=1 such that I d is nowhere dense, dim H I d = θ, and (3.1) 0 ≤ d 1 ≤ d 2 ≤ • • • .Let f n := d n + z n for all n ∈ N.Then, by Lemma 2, we have dim H I q = θ and I f = I d ∪ I z admit a decomposition mentioned in the statement.

Remark 3 .
+ z n )(t)dt ≥ lim n→∞ y x z n (t)dt = +∞ for all x, y ∈ I with x < y, which completes the proof.The set I q in Theorem 1 cannot have positive measure (compare with Proposition 1 and Proposition 2).It turns out that the set D ∞ obtained in Lemma 5 has zero Lebesgue measure.In fact all Borel sets D with dim H D ∈ (0, 1) have zero measure.This is because the Hausdorff measure H d is up to a constant equivalent to the Lebesgue one, i.e. for integer d, H d (D) = c(d)λ d (D) where c(d) is a known constant.Hence, if λ 1 (D) > 0, then H 1 (D) > 0 and thus dim H D = 1.

Theorem 2 .
For every interval I and every θ ∈ [0, 1] there exists a max-family (f n : I → R) ∞ n=1 of continuous functions such that dim H (I f ∩ U) = θ for every open subinterval U of I.