c-ALGEBRABILITY OF PATHOLOGICAL SETS OF PRODUCT INTEGRABLE FUNCTIONS

In this paper we investigate linear algebraic structures in the set of product integrable matrixvalued functions and find c-generated algebras in L([a, b],Rn×n)\L∗([a, b],Rn×n) and D([a, b],Rn×n)\L([a, b],Rn×n).


Introduction
If X is a vector space, a subset M of X is called lineable if M ∪ {0} contains an infinite dimensional vector space.If X is a linear algebra and M ⊆ X, one calls M a κ-algebrable set if M ∪ {0} contains a κ-generated algebra, that is, an algebra which has a minimal system of generators of cardinality κ.These notions were coined by V.I.Guariy [1,9] and then became a criterion for measuring how much large linear algebraic structures could be found in a set of functions with weird properties (see [2,[6][7][8]).
Another criterion is the concept of strong algebrability introduced by Glab and Bartoszewicz in [5].Let κ be a cardinal number and X be a linear commutative algebra.A subset M of X is called strongly κ-algebrable if M ∪ {0} contains a κ-generated algebra isomorphic to a free algebra.
In this paper we seek a linear algebraic structures in the spaces of product integrable function.The notion of product integral has been introduced by Vito Volterra about the end of the 19th century, who studied linear systems of differential equations where I is the identity matrix, A : [a, b] → R n×n is a given continuous function and W : [a, b] → R n×n is the unknown function (see [17]).Later, Ludwig Schlesinger introduced the definition of the Riemann product integral as follows: Given a tagged partition of an interval [a, b], which is a collection of point-interval pairs , where a = t 0 ≤ t 1 ≤ ... ≤ t m = b and ξ i ∈ [t i−1 , t i ] for every i ∈ {1, 2, ..., m}.We refer to t 0 , t 1 , ..., t m as the division points of D, while ξ 1 , ξ 2 , ..., ξ m are the tags of D. In this paper R([a, b], R n×n ) denotes the set of all Riemann product integrable functions.

Now consider a matrix function
Utilizing step functions Schlesinger generalized this definition and introduced the Lebesgue product integral (see [11,12,16]).Let us recall some facts that will be needed: 2. For A ∈ R n×n we will use the operator norm A = sup { Ax : x ≤ 1} , where Ax and x denote the Euclidean norms of vectors Ax, x ∈ R n .After that Schlesinger extended the definition of L * ([a, b], R n×n ) to all matrix functions with Lebesgue integrable (not necessarily bounded) entries and used the next symbole:

A sequence of functions {A
The symbole (L) estands for the Lebesgue integral.Taking account of Theorem 1.1 it is natural to state the following definition.
We define Concerning the above definitions of product integral we have the following chain of strict inclusions:

The exponential function and the product integral
Recall that for every A ∈ R n×n the matrix exponential is defined by where partitions are as in introduction.
. Now every function A k is associated with a partition and So by the definition of Lebesgue product integrable functions, Moreover Schlesinger in [16, p. 485-486] proved the product integral might be also calculated as We remark that each

Lebesgue product integrable functions
The next definition and theorem provide important tools for proving the existence of infinitely generated algebras in the family of real or complex functions.

Definition 3.1 ( [3]
).We say that a function f : R → R is an exponential-like function (of rank m) whenever a i e bix for some distinct nonzero real numbers b 1 , b 2 , ..., b m and some nonzero real numbers a 1 , a 2 , ..., a m . 1]and assume that there exists a function F ∈ F such that f oF ∈ F\{0} for every exponential-like function f : R → R. Then F is strongly c-algebrable.More exactly, if H ⊂ R is a set of cardinality c and linearly independent over the rationals Q, then exp •(rF ), r ∈ H, are free generators of an algebra contained in F ∪ {0}.
Note that in all proofs we apply Theorem 3.2 Theorem 3.3.The set of Riemann real valued integrable functions is strongly c-algebrable.
Proof.Volterra in [17] showed that the Riemann integrable functions are product integrable, thus by Theorem 2.1 and Theorem 3.2 the proof follows.
Theorem 3.4.The set of real valued Lebesgue integrable functions is strongly c-algebrable.
Proof.Schlesinger in [12,16] showed the product integrability of Lebesgue integrable functions.So by Theorem 2.2 and Theorem 3.2, the proof is complete.
Proof.Let A : [0, 1] → R n×n be given by A(x) = (a ij (x)) n i,j=1 such that for each i, j = 1, 2, .., n, So for some y ∈ R n×1 and y ≤ 1, Thus A is not bounded and so A and exp .
Given an arbitrary i and j, and note that for m ≥ 2, , so by the Monotone Convergence Theorem a ij (x) is Lebesgue integrable.Thus A and exp

Product integrability of Denjoy integrable matrix-valued functions
The following definition generalizes the concept of Denjoy product integration.
The abbreviations AC, BV and ACG stand for "absolutely continuous", "bounded variations" and "generalized absolutely continiuous", respectively.1.The strong variation of F on E is defined by where the supremum is taken over all finite collections {[c i , d i ] : 1 ≤ i ≤ n} of non-overlapping intervals that have endpoints in E.

The function A is absolutely continuous in the restricted sense on E (briefely
for each ε > 0, there exists δ > 0 such that  ( ( ( be a perfect set.Then there is a perfect portion ( Note that in this case, each subinterval of [a, b] contains an interval on which the function F is AC(AC * ).
The endpoints of all the intervals on which F is AC(AC * ) form a dence set in [a, b] ).
We recall that the next Lemma and proposition are mentioned in [15] as exercises.for each x ∈ (a n , b n ).In particular, if c is two-sided limit point of E and Proof.First we note that G = F on E and G is linear on each of the intervals contiguous to E. For each x ∈ [a n , b n ], we have and hence an easy calculation completes the proof.
A(ξ i )∆t i ) = (I + A(ξ 1 )∆t 1 )(I + A(ξ 2 )∆t 2 )....(I + A(ξ m )∆t m ).In case the limit lim υ(D)→0 P (A, D) exists, it is called the Riemann product integral of the function A on the interval [a, b] and is denoted by the symbol (I + A(t)dt) b a .

Theorem 1 . 1 ..
[16, Lemma 3.5.4and Theorem 3.5.5]Let A k : [a, b] → R n×n , k ∈ N, be a uniformly bounded sequence of step functions such that lim k→∞ A k (x) = A(x) a.e. on [a, b].Then lim k→∞ A k − A 1 = lim k→∞ b a A k (x) − A(x) dx = 0, and the limit lim k→∞ (I + A k (x)dx) b a exists and is independent of the choice of the sequence {A k }.Definition 1.2.[16, Definiton 3.5.6]Consider the function A : [a, b] → R n×n .Assume there exists a uniformly bounded sequence of step functions A k : [a, b] → R n×n such that lim k→∞ A k (x) = A(x) a.e. on [a, b], then the function A is called Lebesgue product integrable and we define (I + A(x)dx) A k (x)dx) b a The symbole L * ([a, b], R n×n ) denotes the set of all Lebesgue product integrable functions.It is easy to show that e. on [a, b] and lim k→∞ A k − A 1 = 0. Thus by Theorem 1.1 and Definition 1.2, each A ∈ L * ([a, b], R n×n ) is Bochner intagrable.

Remark 1 . 2 .
Since step functions belong to the complete space L([a, b], R n×n ), every product integrable function also belongs to L([a, b], R n×n ).Moreover, step functions form a dense subset in this space, and hence (I + A(x)dx) b a exists if and only if A ∈ L([a, b], R n×n ), i.e., the Lebesgue integral b a A(t) dt is finite.

.
Bochner integrable and hence the product integrals b a exp(A(t)dt and b a (I + A(t)dt) exist and equal to each other; see [13, Theorem 14, Theorem 16].Thus according to the previous discussion, Theorem 2.1 holds for all A ∈ L * ([a, b], R n×n ).Now cosider a function A ∈ L([a, b], R n×n ).By the definition 1.3 there exists a sequence of step functions {A k } ∞ k=1 such that lim k→∞ A k − A 1 = 0 and (I + A(t)dt) A k (t)dt) b a Thus Theorem 2.1 does also hold for A ∈ L([a, b], R n×n ).So we can state the next theorem.Theorem 2.2.Let A : [a, b] → R n×n be a matrix function and A ∈ L([a, b], R n×n ), then exp •A is product integrable.

Definition 4 . 1 .
Consider the function A : [a, b] → R n×n and let [c, d] ⊂ [a, b].The oscilation of A on the interval [c, d] is the number

Definition 4 . 3 .
The function A : [a, b] → R n×n is Denjoy integrable on [a, b] if there exists an ACG * function A : [a, b] → R n×n such that A = A a.e. on [a, b].Theorem 4.4.[15, Theorem 6.2] Let F : [a, b] → R n×n and E ⊆ [a, b].

)
Suppose that E is closed with a, b ∈ E and let G be the linear extension of F to [a, b].If F is BV (AC) on E, then G is BV (AC) on [a, b].Remark 4.1.Let P be a perfect set.A perfect portion of P is a set of the form P ∩[c, d] where P ∩(c, d) = ∅, c, d ∈ P, and P ∩ [c, d] is a perfect set.Theorem 4.5.[15, Theorem 6.10] Suppose that F : [a, b] → R n×n is ACG(ACG * ) on [a, b] and let E ⊂ [a, b]

Lemma 4 . 1 .
Let F : [a, b] → R n×n , and E be a closed set with bounds a and b, and let [a, b] − E = ∞ n=1 (a n , b n ).Suppose that G is the linear extension of F from E to [a, b] and c ∈ E. Then G(x)−G(c) x−c is between F (an)−F (c) an−c and F (bn)−F (c) bn−c

Proposition 4 . 1 .
Suppose that A : [a, b] → R n×n is Denjoy integrable on [a, b].Then [a, b] = ∪ ∞ n=1 E n where each E n is closed and A is Lebesgue integrable on each E n .Proof.By the hypothesis, there exists an ACG * function A : [a, b] → R n×n such that A = A a.e. on [a, b], and we can write [a, b] = ∪ ∞ n=1 E n , where A is AC * on each E n .By Theorem 4.4 we can assume that each E n is closed.Then by Theorem 4.5 there exists a perfect portionE n ∩ [c, d] of E n for n ∈ N, such that A is AC * on E n ∩ [c, d].Let G : [c, d] → R n×nbe the linear extension of A En∩[c,d] to [c, d].By part 3 of Theorem 4.4, G is AC on [c, d].So the function G exists a.e. and is Lebesgue integrable on [c, d].But by Lemma 4.1A = G = A a.e. on E n ∩ [c, d], so the function A is Lebesgue integrable.Theorem 4.6.Let A : [a, b] → R n×n be Denjoy integrable on [a, b], then it is product integrable.Proof.Let D([a, b], R n×n) be endowed by the norm A = (D) b a A(t) dt, where (D) stands for the Denjoy integral.By Proposition 4.1 there exists subsets E n such that [a, b] = ∪ ∞ n=1 E n where for each n ∈ N, E n is non-overlapping, closed and A is Lebesgue integrable on E n .Let A n be the restriction of A to E n for each n ∈ N. Then each A n is Lebesgue integrable and so product integrable and hence for each A n there exists a sequense of step functions{A n k } ∞ k=1 such that A n k : E n → R n×n and lim k→∞ A n k − A n En = lim k→∞ En A n k (x) − A n (x) dx = 0For each n, put a n = inf E n and b n = supE n , so both a n , b n are in E n .Thus for each E n there exist t 0 , t 1 , ..., t n such thatt 0 = a n ≤ t 1 ≤ ... ≤ t n = b n ,andA n k is constant on (t k−1 , t k ) for k = 1, . . ., n.Now let {B k } ∞ k=1 be a sequence of step functions on [a, b] such that [a, b] = ∞ n=1 E n and B k = A n k on each E n .Then by Dominated Convergence Theorem we have the followings:lim k→∞ B k − A 1 = lim k→∞ b a B k (x) − A(x) dx = lim k→∞ ∞ n=1 En A n k (x) − A n (x) dx = 0,i.e., B k converges to A also in the norm of space D([a, b], R n×n ) and hence by [16, Theorem 3.5.5]lim k→∞ (I + B k (x))dx b a exists.So the proof is complete.5. c-algebrability of the set of Denjoy product integrable In this section, some pathological properties (more precisely algebrability) of sets of product integrable functions contained in D([a, b], R n×n )\L([a, b], R n×n ) are investigated.First we note that a matrix A = {a ij } n i,j=1 is called regular if it has a nonzero determinant.Definition 5.1.A function A : [a, b] → R n×n is called Perron product integrable if there is a regular matrix B ∈ R n×n such that for every ε > 0 there is a function δ : [a, b] → (0, ∞) such that B − P (A, D) < ε for every δ-fine partition D of [a,b].Theorem 5.2.Consider the function A : [a, b] → R n×n in D([a, b], R n×n ).Then
The function A is generalized absolutely continuous in the restricted sense on E (briefely A is ACG * on E ) if A E is continuous on E and E can be written as a countable union sets on each of which A is AC * .Note that in general, V (F, E) ≤ V * (F, E) and hence A is BV (AC, BV G, ACG) on E if it is BV * (AC * , BV G * , ACG * ) on E.