ON THE C-INTEGRAL

BENEDETTO BONGIORNO

Let F: [a, b] → ℝ be a differentiable function and let f be its derivative. The problem of recovering F from f is called problem of primitives. In 1912, the problem of primitives was solved by A. Denjoy with an integration process (called totalization) that includes the Lebesgue integral and the Riemann improper integral. Two years later, a second solution was obtained by O. Perron with a method based on the notions of major function and minor function. A third solution, based on a generalization of Riemann integral, is due to J. Kurzweil (1957) and R. Henstock (1963).

It is surprising that, nevertheless the three integration processes are completely different, they produce the same integral (i.e. they have the same space of integrable functions and satisfy the same properties).

In 1986, A.M. Bruckner, R.J. Fleissner and J. Foran [9] remarked that the solution provided by Denjoy, Perron, Kurzweil and Henstock possesses a generality which is not needed for this purpose. In fact the function

(1)  F (x) = \{ \begin{matrix}x sin \frac{1}{x^{2}} & , 0 < x ≤ 1 \\ 0 & , x = 0 \end{matrix}

is a primitive for the Denjoy-Perron-Kurzweil-Henstock integral (more precisely, F is a primitive for the Riemann improper integral), but it is neither a Lebesgue primitive, neither a differentiable function, nor a sum of a Lebesgue primitive and a differentiable function (see [9] for details).

The question of providing a minimal constructive integration process which includes the Lebesgue integral and also integrates the derivatives of differentiable functions was solved by the following Riemann-type integral:

Supported by MURST of Italy.

Definition 1. Given a function f: [a, b] → R we say that f is Cintegrable on [a, b] if there exists a constant A such that for each ε > 0 there is a gauge δ on [a, b] with

(2)  |∑_{i=1}^{p} f(x_{i})|I_{i}| − A| < ε ,

for each δ-fine McShane-partition {(I_{1}, x_{1}), …, (I_{p}, x_{p})} of [a,b] satisfying the condition

(3)  ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε.

Here by “gauge” on [a, b] we mean a positive function δ defined on [a, b], and by “δ-fine McShane-partition of [a, b]” we mean a collection {(I_{1}, x_{1}), …, (I_{p}, x_{p})} of pairwise nonoverlapping intervals I_{i} ⊂ [a, b] and points x_{i} ∈ [a, b] such that I_{i} ⊂ (x_{i}−δ(x_{i}), x_{i}+δ(x_{i})) and ∑_{i=1}^{p} |I_{i}| = b − a.

The number A is called the C-integral of f on [a, b], and we set A = (C) ∫_{a}^{b} f.

The mentioned minimality of the C-integral as a constructive integration process which includes the Lebesgue integral and also integrates the derivatives of differentiable functions follows by the following theorem. Theorem 2. ([5, Main Theorem]) A function f: [a, b] → R is C-integrable on [a, b] if and only if there exists a derivative h such that f − h is Lebesgue integrable.

1. Relations between the C-integral and the Lebesgue integral, the Henstock-Kurzweil integral, and the Riemann-improper integral

It is well known that the Lebesgue integral is equivalent to the McShane integral ([16], [17]); consequently, by Definition 1 it follows that each Lebesgue integrable function is C-integrable with the same value of the integral.

Notice also that if f is C-integrable on [a, b], then f is Henstock-Kurzweil integrable on [a, b] with the same value of the integral. In fact, condition (2) holds for each collection {(I_{1}, x_{1}), …, (I_{p}, x_{p})} of pairwise nonoverlapping intervals I_{i} ⊂ [a, b] and points x_{i} ∈ [a, b] such that I_{i} ⊂ (x_{i} − δ(x_{i}), x_{i} + δ(x_{i})) and x_{i} ∈ I_{i}, i = 1, 2, …, p.

Now we prove that both inclusions are proper.

Proposition 3. The Lebesgue integral is properly contained into the C-integral; i.e. there is a C-integrable function f such that f is not Lebesgue integrable.

Proof. Let us consider the function f on [0, 1] given by

f(x) = 2x sin \frac{1}{x^{2}} − \frac{2}{x} cos \frac{1}{x^{2}} , with f(0) = 0.

A primitive of f is the function F given by F (x) = x^{2} sin(1/x^{2}), F (0) = 0. It is easy to check that F is not absolutely continuous on [0, 1]; hence f is not Lebesgue integrable on [0, 1]. Now remark that F is differentiable everywhere in [0, 1] with derivative F′(x) = f(x), x ∈ [0, 1]. So f is C-integrable, by the following proposition.

Proposition 4. Each derivative is C-integrable.

Proof. Let F be an everywhere differentiable function on [a, b], and let f(x) = F′(x) for each x ∈ [a, b]. Given 0 < ε < 1/(b − a) and x ∈ [a, b], by definition of derivative, there exists δ(x) > 0 such that

(4)  |\frac{F (y) − F (x)}{y − x} − f(x)| < \frac{ε^{2}}{4} ,

for each y ∈ [a, b] with |y − x| < δ(x). Given an interval I = (α, β) we set F (I) = F (β) − F (α) and |I| = β − α. If I is a subinterval of (x − δ(x), x + δ(x)), then by (4) we have

| F (I) − f(x)|I||
≤ |F (β) − F (x) − f(x)(β − x)|

+|F (α) − F (x) − f(x)(α − x)| (5)

< \frac{ε^{2}}{4} |β − x| + \frac{ε^{2}}{4} |α − x|

≤ \frac{ε^{2}}{2} (dist(x, I) + |I|).

Therefore, if {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} is a δ-fine McShane-partition of [a, b] satisfying the condition ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε, then by (5) we get

|∑_{i=1}^{p} f(x_{i})|I_{i}| − (F (b) − F (a))|

≤ ∑_{i=1}^{p} |f(x_{i})|I_{i}| − F (I_{i})| < \frac{ε^{2}}{2} ∑_{i=1}^{p}(dist(x_{i}, I_{i}) + |I_{i}|)

< \frac{ε}{2} + \frac{ε}{2} = ε.

Thus f is C-integrable on [a, b]. □

Proposition 5. The C-integral is properly contained into the Henstock-Kurzweil integral; i.e. there is a Henstock-Kurzweil integrable function f such that f is not C-integrable.

Proof. Let us consider the function (1) and define f = F′ in (0, 1] and f(0) = 0. It is easy to see that f is Riemann improper integrable on [0, 1], hence f is Henstock-Kurzweil integrable. To prove that f is not C-integrable we need the following extension of classical Henstock’s lemma:

Lemma 6. If a function f: [a, b] → ℝ is C-integrable on [a, b], then given ε > 0 there exists a gauge δ on [a, b] so that

(6)  ∑_{i=1}^{p} |f(x_{i})|I_{i}| − (C) ∫_{I_{i}} f| < ε ,

for each δ-fine partial McShane-partition {(I_{1}, x_{1}), …, (I_{p}, x_{p})} in [a, b] with ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε.

Here by “δ-fine partial McShane-partition {(I_{1}, x_{1}), …, (I_{p}, x_{p})} in [a, b]” we mean a collection {(I_{1}, x_{1}), …, (I_{p}, x_{p})} of pairwise nonoverlapping intervals I_{i} ⊂ [a, b] and points x_{i} ∈ [a, b] such that I_{i} ⊂ (x_{i} − δ(x_{i}), x_{i} + δ(x_{i})) and ∑_{i=1}^{p} |I_{i}| < b − a.

The proof of this lemma follows easily by a simple adaptation of standard technics (see [13, Lemma 9.11]).

Assume, by contradiction, that f is C-integrable on [0, 1]. Given ε > 0, take a gauge δ according with Lemma 6, and let a_{h} = (π+2hπ)^{−1/2}, b_{h} = (π/2+2hπ)^{−1/2}. It is easy to see that ∑_{h} a_{h} = ∑_{h} b_{h} = ∞, and that the intervals (a_{h}, b_{h}), h = 1, 2, …, are pairwise disjoint. Moreover, by (1) we have F (a_{h}) = 0, F (b_{h}) = b_{h}, for each h ∈ ℕ. Take natural numbers n, p such that

(a_{n+i}, b_{n+i}) ⊂ (0, δ(0)), i = 1, ⋯, p , and ε < ∑_{1}^{p} a_{n+i} < \frac{1}{ε}.

Hence ∑_{1}^{p} b_{n+i} > ∑_{1}^{p} a_{n+i} > ε. Now define I_{1} = (a_{n+1}, b_{n+1}), ⋯, I_{p} = (a_{n+p}, b_{n+p}). Then {(I_{1}, 0), …, (I_{p}, 0)} is a δ-fine partial McShane-partition in [0, 1], and ∑_{i=1}^{p} dist(0, I_{i}) = ∑_{i=1}^{p} a_{i} < 1/ε. Moreover

∑_{i=1}^{p} |f(0)|I_{i}| − (C) ∫_{I_{i}} f| = ∑_{i=1}^{p} |F (b_{n+i}) − F (a_{n+i})| = ∑_{i=1}^{p} b_{n+i} > ε,

in contradiction with (6). Thus f is not C-integrable on [0, 1]. □

Remark, moreover, that the function (1) is Riemann improper integrable on [0, 1]. Hence next proposition is also proved. Proposition 7. The C-integral doesn’t contain the Riemann improper integral.

2. The variational measure V_{C}F

A first descriptive characterization of the C-primitives was obtained by A.M. Bruckner et al. [9]. They proved that a function F is a C-primitive if and only if F is the limit in variation of a sequence of absolutely continuous functions.

Note that, in general, F is not a function of bounded variation; however, its associate variational measure is absolutely continuous with respect to the Lebesgue measure. This follows from the observation that a C-primitive is also a Henstock-Kurzweil primitive and by [6, Theorem 3].

In fact the last mentioned theorem gives a characterization of Henstock-Kurzweil primitives. The idea of considering appropriate variational measures to characterize the primitives of some integral has been also used in [4, 10, 11] for many multidimensional integrals and in [7] for the Henstock-dyadic integral and for the Henstock-symmetric integral.

Given a function F on [a, b], a gauge δ on [a, b], a subset E of [a, b], and a positive ε, we denote by V_{ε}(F, δ, E) the supremum of all sums ∑_{i} |F (I_{i})| where {(I_{1}, x_{1}), …, (I_{p}, x_{p})} runs into the class of δ-fine partial McShane-partitions of [a, b] with x_{i} ∈ E, i = 1, 2, ⋯ p, and ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε.

It is clear that

(7)  V_{ε_{1}}(F, δ, E) ≤ V_{ε_{2}}(F, δ, E), for ε_{1} > ε_{2}.

Then we define

V_{C}F (E) = sup_{ε} inf_{δ} V_{ε}(F, δ, E).

By the same argument used in [20] for proving Theorem 3.7 and Theorem 3.15, we can show that the extended real-valued set function V_{C}F is a Borel regular measure in [a, b]. Theorem 8. [2, Theorem 4.1] F is an indefinite C-integral if and only if the variational measure V_{C}F is absolutely continuous with respect to the Lebesgue measure.

This characterization is used to prove the following theorem. Theorem 9. [2, Theorem 4.2] Every BV function is a multiplier for the C-integral; i.e. if g is a BV function on [a, b] and f is C-integrable on [a, b], then fg is also C-integrable on [a, b].

Here by BV function we mean a function g: [a, b] → ℝ such that there exists a function of bounded variation g̃: [a, b] → ℝ with g = g̃ almost everywhere in [a, b].

It is known that the product of two derivatives may be not a derivative. A simple consequence of Theorem 9 show that Theorem 10. [2, Theorem 4.3] The product between a derivative and a BV function is a derivative modulo a Lebesgue integrable function of arbitrarily small L^{1}-norm; i.e. if f is a derivative and g is a BV function, then there is a sequence {ϕ_{n}} of derivatives such that ∥fg − ϕ_{n}∥_{1} → 0.

3. ACG_{C} Functions

Definition 11. A function F: [a, b] → ℝ is said to be C-absolutely continuous on a set E ⊂ [a, b] (br. AC_{C}(E)) whenever for each ε > 0 there exist a constant η > 0 and a gauge δ on E such that

(8)  ∑_{i} |F (I_{i})| < ε

for each δ-fine partial McShane-partition {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} in [a, b] satisfying the following conditions:

(α_{1}) x_{i} ∈ E, i = 1, 2, …, p;

(α_{2}) ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε;

(α_{3}) ∑_{i} |I_{i}| < η.

Definition 12. A function F: [a, b] → ℝ is said to be C-generalized absolutely continuous on [a, b] (br. ACG_{C}[a, b]) whenever

(β) there exist measurable sets E_{1}, E_{2}, … such that [a, b] = ⋃_{n} E_{n} and F is AC_{C}(E_{n}), n = 1, 2, ….

According with R.A. Gordon [12] a function F: [a, b] → ℝ is said to be AC_{δ}(E) whenever condition (8) is satisfied for each δ-fine partial McShane-partition {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} satisfying conditions (α_{1}), (α_{3}), and

(α_{2} ′) ∑_{i=1}^{p} dist(x_{i}, I_{i}) = 0.

Moreover F is said to be ACG_{δ}[a, b] whenever

(β′) there exist measurable sets E_{1}, E_{2}, … such that [a, b] = ⋃_{n} E_{n} and F is AC_{δ}(E_{n}), n = 1, 2, . . ..

Therefore each AC_{C}-function is AC_{δ}, and each ACG_{C}-function is ACG_{δ}.

Lemma 13. If F is ACG_{C}[a, b], and E ⊂ [a, b] with |E| = 0, then for each ε > 0 there is a gauge δ on E such that

∑_{i=1}^{p} |F (I_{i})| < ε,

for each δ-fine partial McShane-partition {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} in [a, b] satisfying the following conditions

(α_{1}) x_{i} ∈ E, i = 1, 2, …, p; (α_{2}) ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε.

Theorem 14. A function F is ACG_{C}[a, b] if and only if there is a C-integrable on [a, b] function f such that

(9) F (x) − F (a) = (C)∫_{a}^{x} f(t) dt, for each x ∈ [a, b].

Proof. Assume that F is ACG_{C}[a, b]. Since F is ACG_{δ}[a, b], by [12, Theorem 6] and by [19, Chapter VII, Theorem 7.2], F is differentiable almost everywhere in [a, b]. Let E be the set of points x ∈ [a, b] such that F is not differentiable at x. Then |E| = 0. So, by Lemma 13, given 0 < ε ≤ 1/(b − a) there is a gauge τ on [a, b] such that

∑_{i=1}^{p} |F (I_{i})| < \frac{ε}{4},

for each τ-fine partial McShane-partition {(J_{1}, x_{1}), ⋯, (J_{p}, x_{p})} in [a, b] with ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε and x_{i} ∈ E, i = 1, 2, . . ..

If F is differentiable at y then, by (5) we can find a positive constant γ(y) such that

| F (I) − F′(y)|I| | < \frac{ε^{2}}{2} (dist(y, I) + |I|),

for each interval I ⊂ (y − γ(y), y + γ(y)). Define

δ(y) = \{ \begin{matrix}τ(y) & if y ∈ E, \\ γ(y) & if y ∉ E, \end{matrix}

and

f(y) = \{ \begin{matrix}0 & if y ∈ E, \\ F′(y) & if y ∉ E. \end{matrix}

Given x ∈ [a, b] let {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} be a δ-fine McShanepartition of [a, x] such that ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε. Then we have

|∑_{i=1}^{p} f(x_{i})|I_{i}| − (F (x) − F (a))|

≤ ∑_{i=1}^{p} |f(x_{i})|I_{i}| − F (I_{i})|

< ∑_{x_{i}∈E} |F (I_{i})| + ∑_{x_{i}∉E} |F′(x_{i})|I_{i}| − F (I_{i})|

< \frac{ε}{2} + ∑_{x_{i}∉E} \frac{ε^{2}}{4} (dist(x_{i}, I_{i}) + |I_{i}|)

< \frac{ε}{2} + \frac{ε}{4} + \frac{ε}{4}ε(b − a) ≤ ε.

Thus f is C-integrable on [a, x] and (9) holds. Taking x = b we see that f is C-integrable on [a, b].

Now assume that f is C-integrable on [a, b] and let F be the C-primitive of f. For each natural number n we define E_{n} = {x ∈ [a, b] : |f(x)| ≤ n}. Then [a, b] = ⋃_{n} E_{n}. To complete the proof it is enough to show that F is AC_{C} on E_{n}, for each n. By Lemma 6, given ε > 0 there is a gauge δ on [a, b] such that

∑_{i=1}^{p} |f(x_{i})|I_{i}| − F (I_{i})| < \frac{ε}{2},

for each δ-fine partial McShane-partition {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} in [a, b] with ∑_{i=1}^{p} dist(x_{i}, I_{i}) < 1/ε. Assume that x_{i} ∈ E_{n}, i = 1, 2, …, and ∑_{i} |I_{i}| < ε/2n. Then

∑_{i=1}^{p} |F (I_{i})|

≤ ∑_{i=1}^{p} |f(x_{i})|I_{i}| − F (I_{i})| + ∑_{i=1}^{p} |f(x_{i})| · |I_{i}|

< \frac{ε}{2} + n ∑_{i} |I_{i}| < ε.

So F is AC_{C}(E_{n}), and the proof is complete. □

4. Convergence Theorems.

Let f be Lebesgue (resp. Henstock-Kurzweil) integrable on [a, b]. We denote by (L) ∫_{a}^{b} f (resp. (HK) ∫_{a}^{b} f) the Lebesgue (resp. Henstock-Kurzweil) integral of f on [a, b].

Monotone convergence theorem. Let f_{1} ≤ f_{2} ≤ ⋯ ≤ f_{n} ⋯ be a sequence of C-integrable functions on [a, b] such that the limit lim_{n}(C)∫_{a}^{b} f_{n} exists finite. Then the function f(x) = lim_{n} f_{n}(x) is C-integrable on [a, b] and we have

(C)∫_{a}^{b} f(x) dx = lim_{n→∞}(C)∫_{a}^{b} f_{n}(x) dx.

Proof. Since each C-integrable function is Henstock-Kurzweil integrable, by [18, Corollary 6.3.5] each nonnegative C-integrable function is Lebesgue integrable, and by [18, Theorem 6.3.3] each C-integrable function is Lebesgue measurable. Then we can apply the Lebesgue monotone convergence theorem to the following sequence

0 ≤ f_{2} − f_{1} ≤ f_{3} − f_{1} ≤ ⋯ f_{n} − f_{1} ≤ ⋯

Hence

(10) (L)∫_{a}^{b}{f(x) − f_{1}(x)} dx = lim_{n→∞}(L)∫_{a}^{b}{f_{n}(x) − f_{1}(x)} dx.

Now for each n ∈ ℕ we have

(L)∫_{a}^{b}{f_{n}(x) − f_{1}(x)} dx
= (C)∫_{a}^{b}{f_{n}(x) − f_{1}(x)} dx
= (C)∫_{a}^{b} f_{n}(x) dx − (C)∫_{a}^{b} f_{1}(x) dx.

Therefore, since lim_{n}(C)∫_{a}^{b} f_{n} exists finite, we can use (10) to infer that f − f_{1} is Lebesgue integrable; hence f = (f − f_{1}) + f_{1} is C-integrable. Moreover

(C)∫_{a}^{b} f(x) dx

= (C)∫_{a}^{b}{f(x) − f_{1}(x)} dx + (C)∫_{a}^{b} f_{1}(x) dx

= lim_{n→∞}(C)∫_{a}^{b} f_{n}(x) dx − (C)∫_{a}^{b} f_{1}(x) dx + (C)∫_{a}^{b} f_{1}(x) dx

= lim_{n→∞}(C)∫_{a}^{b} f_{n}(x) dx.

□

Dominated convergence theorem. Let f_{1}, f_{2}, ⋯, f_{n} ⋯ be a sequence of measurable functions such that

(i) f_{n}(x) → f(x) almost everywhere in [a, b]; (ii) g(x) ≤ f_{n}(x) ≤ h(x), almost everywhere in [a, b], with g

and h C-integrable on [a, b];

then f is C-integrable on [a, b] and

(C)∫_{a}^{b} f(x) dx = lim_{n→∞}(C)∫_{a}^{b} f_{n}(x) dx.

Proof. By (ii) we have 0 ≤ f_{n} − g ≤ h − g, almost everywhere in [a, b]. Moreover h−g is Lebesgue integrable, since non negative and C-integrable. Then, by Lebesgue dominated convergence theorem, f − g is Lebesgue integrable on [a, b] with

(L)∫_{a}^{b}{f(x) − g(x)} dx = lim_{n→∞}(L)∫_{a}^{b}{f_{n}(x) − g(x)} dx.

Therefore, by f = (f − g) + g we infer that f is C-integrable on [a, b] with

(C)∫_{a}^{b} f(x) dx = (C)∫_{a}^{b}{f(x) − g(x)} dx + (C)∫_{a}^{b} g(x) dx

= (L)∫_{a}^{b}{f(x) − g(x)} dx + (C)∫_{a}^{b} g(x) dx

= lim_{n→∞}(C)∫_{a}^{b} f_{n}(x) dx − (C)∫_{a}^{b} g(x) dx + (C)∫_{a}^{b} g(x) dx

= lim_{n→∞}(C)∫_{a}^{b} f_{n}(x) dx.

□

Definition 15. Let E be a subset of [a, b] and let {F_{n}} be a sequence of real valued functions defined on [a, b]. It is said that {F_{n}} is uniformly AC_{C}(E) if for each ε > 0 there exist a constant η > 0 and a gauge δ such that

sup_{n} ∑_{i} |F_{n}(I_{i})| < ε,

for each δ-fine partial McShane-partition {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} in [a, b], satisfying conditions (α_{1}), (α_{2}), (α_{3}). Definition 16. A sequence {F_{n}} is said to be uniformly ACG_{C}[a, b] if there exist measurable sets E_{1}, E_{2}, … such that [a, b] = ⋃_{k} E_{k} and {F_{n}} is uniformly AC_{C}(E_{k}), k = 1, 2, ….

Analogously we define the notions of sequence uniformly AC_{δ}(E) and uniformly ACG_{δ}[a, b]. It is easy to see that if a sequence {F_{n}} is uniformly ACG_{C}[a, b] then {F_{n}} is uniformly ACG_{δ}[a, b].

Controlled convergence theorem. Let {f_{n}} be a sequence of C-integrable on [a, b] functions such that

(i) f_{n}(x) → f(x) almost everywhere in [a, b]; (ii) the sequence F_{n}(x) = (C)∫_{a}^{x} f_{n}(t) dt is uniformly ACG_{C}[a, b].

Then f is C-integrable on [a, b] and

(11)  (C)∫_{a}^{b} f(x) dx = lim_{n→∞}(C)∫_{a}^{b} f_{n}(x) dx.

Proof. Since each C-integrable function is Henstock-Kurzweil integrable and since each uniformly ACG_{C} sequence is uniformly ACG_{δ}, then by [3, Theorem 4.1] f is Henstock-Kurzweil integrable on [a, b] and

(12)  (HK)∫_{a}^{x} f(t) dt = lim_{n→∞}(HK)∫_{a}^{x} f_{n}(t) dt,

for each x ∈ [a, b]. Now, by (ii), there exist a sequence of measurable sets {E_{h}} such that

• [a, b] = ⋃_{h} E_{h},

• for each ε > 0 and each h there exist a constant η_{h} > 0 and a gauge δ_{h} such that

(13)  sup_{n} ∑_{i} |F_{n}(I_{i})| < ε,

for each δ_{h}-fine partial McShane-partition {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} in [a, b], satisfying conditions (α_{1}), (α_{2}), (α_{3}), with E_{h} for E.

Set F (x) = (HK)∫_{a}^{x} f(t) dt. Since

F_{n}(x) = (C)∫_{a}^{x} f_{n}(t) dt = (HK)∫_{a}^{x} f_{n}(t) dt,

by (12) we have F (x) = lim_{n} F_{n}(x), for each x ∈ [a, b]. Consequently, by (13), for each h ∈ ℕ and for each δ_{h}-fine partial McShane-partition {(I_{1}, x_{1}), ⋯, (I_{p}, x_{p})} in [a, b] satisfying conditions (α_{1}), (α_{2}), (α_{3}), with E_{h} for E, we have

∑_{i=1}^{p} |F (I_{i})| = lim_{n→∞} ∑_{i=1}^{p} |F_{n}(I_{i})| ≤ ε.

Thus F is ACG_{C}[a, b] and by Theorem 14, F (x) = (C)∫_{a}^{x} f(t) dt for each x ∈ [a, b]. In conclusion (11) follows easily by (12). □

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Department of Mathematics, University of Palerme,, Via Archirafi 34, 90123 Palermo (Italy)

  E-mail address: bb@math.unipa.it