International Journal of Analysis and Applications Volume 16, Number 5 (2018), 702-711 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-702 c-ALGEBRABILITY OF PATHOLOGICAL SETS OF PRODUCT INTEGRABLE FUNCTIONS FATEMEH FARMANESH, ALI FAROKHINIA∗ Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran ∗Corresponding author: alifarokinia@gmail.com Abstract. In this paper we investigate linear algebraic structures in the set of product integrable matrix- valued 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). 1. 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, 6–8]). 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 Received 2018-04-26; accepted 2018-08-02; published 2018-09-05. 2000 Mathematics Subject Classification. 47A16, 47L10. Key words and phrases. algebrable; Lebesgue integrable; product integrable. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 702 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-702 Int. J. Anal. Appl. 16 (5) (2018) 703 linear systems of differential equations W ′(t) = A(t)W(t), t ∈ [a,b] W(a) = I, where I is the identity matrix, A : [a,b] → Rn×n is a given continuous function and W : [a,b] → Rn×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 D = (ξi, [ti−1, ti]) m i=1, where a = t0 ≤ t1 ≤ ... ≤ tm = b and ξi ∈ [ti−1, ti] for every i ∈{1, 2, ...,m}. We refer to t0, t1, ..., tm as the division points of D, while ξ1,ξ2, ...,ξm are the tags of D. Remark 1.1. If we replace ξi ∈ [ti−1, ti] by ξi ∈ [a,b], then the collection D is called a free tagged partition. Given a function δ : [a,b] → R+ (called a gauge on [a,b]), a free tagged partition is called δ-fine if [ti−1, ti] ⊂ (ξi −δ(ξi),ξi + δ(ξi)) , i = {1, 2, ...,m}. Now consider a matrix function A : [a,b] → Rn×n with entries {aij} n i,j=1 . Put ∆ti = ti − ti−1 , i = 1, 2, ...,m , υ(D) = max ∆ti 1≤i≤m , and define P(A,D) = m∏ i=1 (I + A(ξi)∆ti) = (I + A(ξ1)∆t1)(I + A(ξ2)∆t2)....(I + A(ξm)∆tm). In case the limit limυ(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 . In this paper R([a,b],Rn×n) denotes the set of all Riemann product integrable functions. Utilizing step functions Schlesinger generalized this definition and introduced the Lebesgue product inte- gral (see [11, 12, 16]). Let us recall some facts that will be needed: 1. A function A : [a,b] → Rn×n is called a step function if there exists numbers a = t0 < t1 < ... < tm = b such that A is constant function on every interval (tk−1, tk), k = 1, 2, ...,m. 2. For A ∈ Rn×n we will use the operator norm ‖A‖ = sup{‖Ax‖ : ‖x‖≤ 1} , where ‖Ax‖ and ‖x‖ denote the Euclidean norms of vectors Ax, x ∈ Rn. 3. A sequence of functions {Ak : [a,b] → Rn×n}k∈N is called uniformly bounded if there exists a number M ∈ R such that ‖Ak(x)‖≤ M for all k ∈ N and all x ∈ [a,b]. Int. J. Anal. Appl. 16 (5) (2018) 704 Theorem 1.1. [16, Lemma 3.5.4 and Theorem 3.5.5] Let Ak : [a,b] → Rn×n, k ∈ N, be a uniformly bounded sequence of step functions such that limk→∞Ak(x) = A(x) a.e. on [a,b]. Then lim k→∞ ‖Ak −A‖1 = lim k→∞ b∫ a ‖Ak(x) −A(x)‖dx = 0, and the limit lim k→∞ (I + Ak(x)dx) b∏ a exists and is independent of the choice of the sequence {Ak}. Definition 1.2. [16, Definiton 3.5.6] Consider the function A : [a,b] → Rn×n. Assume there exists a uniformly bounded sequence of step functions Ak : [a,b] → Rn×n such that limk→∞Ak(x) = A(x) a.e. on [a,b], then the function A is called Lebesgue product integrable and we define (I + A(x)dx) b∏ a = lim k→∞ (I + Ak(x)dx) b∏ a . The symbole L∗([a,b],Rn×n) denotes the set of all Lebesgue product integrable functions. It is easy to show that L∗([a,b],Rn×n) = {A : [a,b] → Rn×n : A is measurable and bounded}. Let us recall that a function A : [a,b] → Rn×n is called Bochner intagrable if there is a sequence of simple functions Ak : [a,b] → Rn×n, k ∈ N such that lim k→∞ Ak(t) = A(t) a.e. on [a,b] and lim k→∞ ‖Ak −A‖1 = 0. Thus by Theorem 1.1 and Definition 1.2, each A ∈ L∗([a,b],Rn×n) is Bochner intagrable. After that Schlesinger extended the definition of L∗([a,b],Rn×n) to all matrix functions with Lebesgue integrable (not necessarily bounded) entries and used the next symbole: L([a,b],Rn×n) = {A : [a,b] → Rn×n : (L) b∫ a ‖A(x)‖dx < ∞}. The symbole (L) estands for the Lebesgue integral. Taking account of Theorem 1.1 it is natural to state the following definition. Definition 1.3. [16, Definiton 3.8.1] A function A : [a,b] → Rn×n is called product integrable if there exists a sequence of step functions {Ak} such that lim k→∞ ‖Ak −A‖1 = 0. We define (I + A(x)dx) b∏ a = lim k→∞ (I + Ak(x)dx) b∏ a Int. J. Anal. Appl. 16 (5) (2018) 705 Remark 1.2. Since step functions belong to the complete space L([a,b],Rn×n), every product integrable function also belongs to L([a,b],Rn×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],Rn×n), i.e., the Lebesgue integral ∫ b a ‖A(t)‖dt is finite. Concerning the above definitions of product integral we have the following chain of strict inclusions: R([a,b],Rn×n) ⊂ L∗([a,b],Rn×n) ⊂ L([a,b],Rn×n). 2. The exponential function and the product integral Recall that for every A ∈ Rn×n the matrix exponential is defined by eA = ∞∑ k=0 Ak k! . Theorem 2.1. [16, Theorem 3.2.2] Consider a Riemann integrable function A : [a,b] → Rn×n. Then lim υ(D)→0 m∏ k=1 eA(ξk)∆tk = lim υ(D)→0 m∏ k=1 (I + A(ξk)∆tk) = (I + A(t)dt) b∏ a , where partitions are as in introduction. Remark 2.1. If A ∈ L∗([a,b],Rn×n) and {Ak} ∞ k=1 is a uniformly bounded sequence of step functions in L∗([a,b],Rn×n) such that Ak → A a.e. on [a,b], then by [16, Theorem 3.6.3] we have (I+A(x)dx) b∏ a = lim k→∞ (I+ Ak(x)dx) b∏ a . Now every function Ak is associated with a partition Dk : a = t k 0 < t k 1 < ... < t k m(k) = b such that Ak(x) = A k j , x ∈ (t k j−1, t k j ), and lim k→∞ υ(Dk) = 0. So by the definition of Lebesgue product integrable functions, (I + A(x)dx) b∏ a = lim k→∞ (I + Ak(x)dx) b∏ a = lim k→∞ m(k)∏ j=1 exp(Akj ∆t k j ). Moreover Schlesinger in [16, p. 485-486] proved the product integral might be also calculated as (I + A(x)dx) b∏ a = lim k→∞ m(k)∏ j=1 (I + (Akj ∆t k j ). We remark that each A ∈ L∗([a,b],Rn×n) is 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],Rn×n). Int. J. Anal. Appl. 16 (5) (2018) 706 Now cosider a function A ∈ L([a,b],Rn×n). By the definition 1.3 there exists a sequence of step functions {Ak} ∞ k=1 such that lim k→∞ ‖Ak −A‖1 = 0 and (I + A(t)dt) b∏ a = lim k→∞ (I + Ak(t)dt) b∏ a . Thus Theorem 2.1 does also hold for A ∈ L([a,b],Rn×n). So we can state the next theorem. Theorem 2.2. Let A : [a,b] → Rn×n be a matrix function and A ∈ L([a,b],Rn×n), then exp◦A is product integrable. 3. 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 f is given by f(x) = m∑ i=1 aie bix for some distinct nonzero real numbers b1,b2, ...,bm and some nonzero real numbers a1,a2, ...,am. Theorem 3.2 ( [3,4]). Let F ⊂ R[0,1] and assume that there exists a function F ∈F such that foF ∈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. � Theorem 3.5. The set L([a,b],Rn×n)\L∗([a,b],Rn×n) is strongly c-algebrable. Proof. Let A : [0, 1] → Rn×n be given by A(x) = (aij(x))ni,j=1 such that for each i,j = 1, 2, ..,n, aij(x) =   1√ x x ∈ (0, 1] 0 x = 0 . Int. J. Anal. Appl. 16 (5) (2018) 707 So for some y ∈ Rn×1 and ‖y‖≤ 1, A(x)y =   a11 . . . a1n ... . . . ... an1 · · · ann     y1 ... yn   =   a11y1 + a12y2 + · · · + a1nyn ... an1y1 + an2y2 + · · · + annyn   , ‖A(x)y‖≥ √ n x (y1 + · · · + yn) 2 ≥ 1 x , x ∈ (0, 1]. Thus A is not bounded and so A and exp◦ (A) are not in L∗([0, 1],Rn×n). Now let Am : [0, 1] → Rn×n be given by Am(x) = ( (m) bij (x)) n i,j=1 such that for each i,j = 1, 2, ..,n, (m) bij (x) =   0 x ∈ [0, 1 m ] 1√ x x ∈ ( 1 m , 1] . Given an arbitrary i and j, and note that for m ≥ 2, (m) bij (x) is Lebesgue integrable on [0, 1]. Since lim m→∞ (m) bij (x) = aij(x) for each x ∈ [0, 1], so by the Monotone Convergence Theorem aij(x) is Lebesgue integrable. Thus A and exp◦(A) are in L([0, 1],Rn×n) so f ◦(A) is in L([0, 1],Rn×n), for every exponential- like function f, and the proof is complete by Theorem 3.2. � Theorem 3.6. The set of L([a,b],Rn×n)\R([a,b],Rn×n) is c-algebrable. Proof. Since L([a,b],Rn×n)\L∗([a,b],Rn×n) ⊆ L([a,b],Rn×n)\R([a,b],Rn×n), the preceding theorem implies that L([a,b],Rn×n)\R([a,b],Rn×n) is c-algebrable. � 4. Product integrability of Denjoy integrable matrix-valued functions The following definition generalizes the concept of Denjoy product integration. Definition 4.1. Consider the function A : [a,b] → Rn×n and let [c,d] ⊂ [a,b]. The oscilation of A on the interval [c,d] is the number osc(A, [c,d]) = sup{‖A(ξ1) −A(ξ2)‖ : ξ1,ξ2 ∈ [c,d]} . The abbreviations AC, BV and ACG stand for “absolutely continuous”, “bounded variations” and “generalized absolutely continiuous”, respectively. Definition 4.2. Let A : [a,b] → Rn×n and E ∈ [a,b]. 1. The strong variation of F on E is defined by V∗(F,E) = sup { n∑ i=1 osc(F, [ci,di]) } , where the supremum is taken over all finite collections {[ci,di] : 1 ≤ i ≤ n} of non-overlapping inter- vals that have endpoints in E. Int. J. Anal. Appl. 16 (5) (2018) 708 2. The function F is of bounded variation in the restricted sense on E (briefely A is BV∗ on E ) if V∗(F,E) is finite. 3. The function A is absolutely continuous in the restricted sense on E (briefely A is AC∗ on E ) if for each ε > 0, there exists δ > 0 such that n∑ i=1 osc(A, [ci,di]) < ε, whenevere {[ci,di] : 1 ≤ i ≤ n} is a finite collection of non-overlapping inter- vals that have endpoints in E and satisfy n∑ i=1 (di − ci) < δ. 4. 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. Definition 4.3. The function A : [a,b] → Rn×n is Denjoy integrable on [a,b] if there exists an ACG∗ function A : [a,b] → Rn×n such that A′ = A a.e. on [a,b]. Theorem 4.4. [15, Theorem 6.2] Let F : [a,b] → Rn×n and E ⊆ [a,b]. (1) If F is AC(ACG,AC∗,ACG∗) on E, then F is BV (BV G,BV∗,BV G∗) on E. (2) If F is BV∗ on E, then F is BV∗ on E. (3) 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) 6= ∅, c,d ∈ P, and P ∩ [c,d] is a perfect set. Theorem 4.5. [15, Theorem 6.10] Suppose that F : [a,b] → Rn×n is ACG(ACG∗) on [a,b] and let E ⊂ [a,b] be a perfect set. Then there is a perfect portion E ∩ [c,d] of E such that F is AC(AC∗) on E ∩ [c,d]. ( 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. Lemma 4.1. Let F : [a,b] → Rn×n, and E be a closed set with bounds a and b, and let [a,b] − E = ∞⋃ n=1 (an,bn). 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 for each x ∈ (an,bn). In particular, if c is two-sided limit point of E and F is differentiable at c, then G is differentiable at c and G′(c) = F ′(c). Int. J. Anal. Appl. 16 (5) (2018) 709 Proof. First we note that G = F on E and G is linear on each of the intervals contiguous to E. For each x ∈ [an,bn], we have G(x) = F(bn) −F(an) bn −an (x−an) + F(an), and hence an easy calculation completes the proof. � Proposition 4.1. Suppose that A : [a,b] → Rn×n is Denjoy integrable on [a,b]. Then [a,b] = ∪∞n=1En where each En is closed and A is Lebesgue integrable on each En. Proof. By the hypothesis, there exists an ACG∗ function A : [a,b] → Rn×n such that A′ = A a.e. on [a,b], and we can write [a,b] = ∪∞n=1En, where A is AC∗ on each En. By Theorem 4.4 we can assume that each En is closed. Then by Theorem 4.5 there exists a perfect portion En ∩ [c,d] of En for n ∈ N, such that A is AC∗ on En∩[c,d]. Let G : [c,d] → Rn×n be 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 4.1 A′ = G′ = A a.e. on En ∩ [c,d], so the function A is Lebesgue integrable. � Theorem 4.6. Let A : [a,b] → Rn×n be Denjoy integrable on [a,b], then it is product integrable. Proof. Let D([a,b],Rn×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 En such that [a,b] = ∪∞n=1En where for each n ∈ N, En is non-overlapping, closed and A is Lebesgue integrable on En. Let An be the restriction of A to En for each n ∈ N. Then each An is Lebesgue integrable and so product integrable and hence for each An there exists a sequense of step functions {Ank} ∞ k=1 such that Ank : En → R n×n and lim k→∞ ‖Ank −An‖En = limk→∞ ∫ En ‖Ank (x) −An(x)‖dx = 0 For each n, put an = infEn and bn = supEn, so both an,bn are in En. Thus for each En there exist t0, t1, ..., tn such that t0 = an ≤ t1 ≤ ... ≤ tn = bn, and Ank is constant on (tk−1, tk) for k = 1, . . . ,n. Now let {Bk} ∞ k=1 be a sequence of step functions on [a,b] such that [a,b] = ∞⋃ n=1 En and Bk = Ank on each En. Then by Dominated Convergence Theorem we have the followings: lim k→∞ ‖Bk −A‖1 = lim k→∞ b∫ a ‖Bk(x) −A(x)‖dx = lim k→∞ ∞∑ n=1 ∫ En ‖Ank (x) −An(x)‖dx = 0, Int. J. Anal. Appl. 16 (5) (2018) 710 i.e., Bk converges to A also in the norm of space D([a,b],Rn×n) and hence by [16, Theorem 3.5.5] lim k→∞ (I + Bk(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],Rn×n)\L([a,b],Rn×n) are investigated. First we note that a matrix A = {aij} n i,j=1 is called regular if it has a nonzero determinant. Definition 5.1. A function A : [a,b] → Rn×n is called Perron product integrable if there is a regular matrix B ∈ Rn×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] → Rn×n in D([a,b],Rn×n). Then b∏ a eA(t)dt = (I + A(t)dt) b∏ a . Proof. By [10, Theorem 2.12] and [15, Theorem 11.2], the proof is clear. � Corollary 5.1. If A : [a,b] → Rn×n is product integrable function, then exp ◦ (A) is product integrable function. Theorem 5.3. The set of product integrable functions is strongly c-algebrable. Proof. By Corollary 5.1 and Theorem 3.2 the proof follows. � Proposition 5.1. [15, Theorem 7.11] Suppose that f : [a,b] → R is Denjoy integrable on each subinterval [c,d] ⊆ (a,b). If d∫ c f converges to a finite limit as c → a+ and d → b−, then f is Denjoy integrable on [a,b] and b∫ a f = lim c→a+ d→b− d∫ c f. Theorem 5.4. The set of D([a,b],Rn×n)\L ([a,b],Rn×n) is strongly c-algebrable. Proof. Let ∞∑ n=1 cn be a nonabsolutely convergent series of real numbers and let In = ( 2−n, 2−n+1 ) , n ∈ N. Define the function A : [0, 1] → Rn×n by A(x) = (aij(x))ni,j=1, such that for each i,j = 1, 2, ...,n (aij(x)) =   2ncn x ∈ In 0 otherwise. Note that 1∫ 0 |aij(x)|dx = ∞∑ n=1 ∫ In |aij(x)|dx = ∞∑ n=1 ∣∣2−ncn2n∣∣ = ∞∑ n=1 |cn| = ∞. Int. J. Anal. Appl. 16 (5) (2018) 711 Hence neither aij nor A is Lebesgue integrable on [0, 1]. Now we are going to show that A is Denjoy integrable on [0, 1]. For each 0 < α < 1 both of functions aij and A are bounded on [α, 1], so they are Lebesgue integrable on [α, 1]. Let B(x) = 1∫ x aij for each x ∈ (0, 1]. The function B is linear on each In. It follows that B(x) is between B(2−n) and B(2n) for each x ∈ In. Now B(2−n) = n∑ k=1 ck and lim n→∞ B (2−n) = ∞∑ k=1 ck. Therefore lim x→0+ B (x) = ∞∑ n=1 cn and according to Proposition 5.1, aij is Denjoy integrable on [0, 1] for each i,j = 1, 2, ...,n. Thus for each aij(x) there exists an ACG∗ function fij such that f ′ ij(x) = aij(x) a.e. on x ∈ [0, 1]. Now put F(x) = (fij(x))ni,j=1 for each x ∈ [0, 1]. So F ′(x) = (f′ij(x)) n i,j=1 = (aij(x)) n i,j=1 = A(x) a.e. on [0,1]. Hence A is Denjoy integrable on x ∈ [0, 1]. One can see easily that exp◦aij is Denjoy integrable and so is exp◦A. Thus by Theorem 3.2 the proof is complete. � References [1] R. Aron, V. I. Gurariy and J. B. Seoane-Sepúlveda, lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc. 133 (3) (2005), 795-803. [2] R. Aron and J. B. Seoane-Sepúlveda, Algebrability of the set of everywhere surjective functions on C, Bull. Belg. Math. Soc. Simon Stevin 14 (1) (2007), 25-31. [3] M. Balcerzak, A. Bartoszewicz and M. Filipczac, Nonseparable spaceability and strong algebrability of sets of continuous singular functions, J. Math. Anal. Appl. 407 (2) (2013), 263-269. [4] A. Bartoszewicz, M. Bieniea, M. Filipczac and S. Glab, Strong c-algebrability of strong Sierpinski-Zygmund, smooth nowhere analytic and other sets of functions, J. Math. Anal. Appl. 412 (2) (2015), 620-630. [5] A. Bartoszewicz and S. Glab, Strong c-algebrability of sets of sequences and functions, Proc. Amer. Math. Soc. 141 (2013), 827-835. [6] A. Farokhinia, Algebrability of space of quasi-everywhere surjective functions. B. Math. Anal. Appl. 6 (6) (2014), 38-43. [7] A. Farokhinia, Lineability of Denjoy integrable function, J. Math. Ext. 11 (1) (2017), 57-65. [8] A. Farokhinia, Lineability of space of quasi-everywhere surjective functions. J. Math. Ext. 6 (3) (2013), 45-51. [9] V. I. Gurariy and L. Qurta, On lineability of sets of continuous functions, J. Math. Anal. Appl. (1) (2004), 62-72. [10] J. Jarńık and J. Kurzweil, A general form of the product integral and linear ordinary, Czech. Math. J. 37 (4) (1987), 642-659. [11] F. R. Riesz, Sur lintegrale de Lebesgue. Acta Mathematica 42 (1919), 191-205. [12] L. Schlesinger, Neue Grandlagen fur einen Infinittesimalkul der Matrizen, Mathematische Zeitschrift 33 (1931), 33-61. [13] A. Slav́ık and S̆. Schwabic, Henstock-Kurzweil and McShane product integrals; descriptive definations, Czech. Math. J. 58 (133) (2008), 241-269. [14] S̆. Schwabic, Bochner product integration, Math. Bohem. 119 (1994), 305-335. [15] R. Gordon, The integrals of Lebesgue, Denjoy, Perron and Henstock, American Mathematical Society, 1994. [16] A. Slav́ık, Product integration, its history and applications, Matfyzpress, Prague, 2007. [17] V. Volterra and B. Hostinsky, Operations infinitesimales lineaires, Gauthier-Villars, Paris, 1938. Adison-Wesley. Publishing Company 1979. 1. Introduction 2. The exponential function and the product integral 3. Lebesgue product integrable functions 4. Product integrability of Denjoy integrable matrix-valued functions 5. c-algebrability of the set of Denjoy product integrable References