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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section THE GH k INTEGRAL FOR RIESZ RIESZ SPACES SPACE-VALUED FUNCTIONS a survey on recent results Antonio Boccuto Department of

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS

a survey on recent results Antonio Boccuto

Department of Mathematics and Computer Sciences University of Perugia Measure Theory Marczewski Centennial Conference Be ¸dlewo, September 10-14, 2007

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Introduction

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Historical survey

REAL-VALUED FUNCTIONS: A. G. DAS, S. KUNDU (2003-2005);

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Historical survey

REAL-VALUED FUNCTIONS: A. G. DAS, S. KUNDU (2003-2005);

  • S. PAL, D. K. GANGULY, P

. Y. LEE (2005);

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Historical survey

REAL-VALUED FUNCTIONS: A. G. DAS, S. KUNDU (2003-2005);

  • S. PAL, D. K. GANGULY, P

. Y. LEE (2005); k = 1: ”generalized Perron integral”, ˇ

  • S. SCHWABIK (1992)
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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Historical survey

REAL-VALUED FUNCTIONS: A. G. DAS, S. KUNDU (2003-2005);

  • S. PAL, D. K. GANGULY, P

. Y. LEE (2005); k = 1: ”generalized Perron integral”, ˇ

  • S. SCHWABIK (1992)

(Generalization of the Kurzweil-Henstock and the Henstock-Stieltjes integrals).

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SLIDE 7

THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Historical survey

REAL-VALUED FUNCTIONS: A. G. DAS, S. KUNDU (2003-2005);

  • S. PAL, D. K. GANGULY, P

. Y. LEE (2005); k = 1: ”generalized Perron integral”, ˇ

  • S. SCHWABIK (1992)

(Generalization of the Kurzweil-Henstock and the Henstock-Stieltjes integrals). METRIC SEMIGROUPS: A. B., D. CANDELORO,

  • A. R. SAMBUCINI (2007)
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SLIDE 8

THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Historical survey

REAL-VALUED FUNCTIONS: A. G. DAS, S. KUNDU (2003-2005);

  • S. PAL, D. K. GANGULY, P

. Y. LEE (2005); k = 1: ”generalized Perron integral”, ˇ

  • S. SCHWABIK (1992)

(Generalization of the Kurzweil-Henstock and the Henstock-Stieltjes integrals). METRIC SEMIGROUPS: A. B., D. CANDELORO,

  • A. R. SAMBUCINI (2007)

GHk-INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS: A. B., B. RIE ˇ CAN, A. R. SAMBUCINI (2007)

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SLIDE 9

THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Historical survey

REAL-VALUED FUNCTIONS: A. G. DAS, S. KUNDU (2003-2005);

  • S. PAL, D. K. GANGULY, P

. Y. LEE (2005); k = 1: ”generalized Perron integral”, ˇ

  • S. SCHWABIK (1992)

(Generalization of the Kurzweil-Henstock and the Henstock-Stieltjes integrals). METRIC SEMIGROUPS: A. B., D. CANDELORO,

  • A. R. SAMBUCINI (2007)

GHk-INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS: A. B., B. RIE ˇ CAN, A. R. SAMBUCINI (2007) Extension Cauchy and convergence theorems

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

Historical survey

REAL-VALUED FUNCTIONS: A. G. DAS, S. KUNDU (2003-2005);

  • S. PAL, D. K. GANGULY, P

. Y. LEE (2005); k = 1: ”generalized Perron integral”, ˇ

  • S. SCHWABIK (1992)

(Generalization of the Kurzweil-Henstock and the Henstock-Stieltjes integrals). METRIC SEMIGROUPS: A. B., D. CANDELORO,

  • A. R. SAMBUCINI (2007)

GHk-INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS: A. B., B. RIE ˇ CAN, A. R. SAMBUCINI (2007) Extension Cauchy and convergence theorems Fundamental Formula of Calculus

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

RIESZ SPACES

A regulator or (D)-sequence is a bounded double sequence (ai,j)i,j in R s. t. ai,j ↓ 0 ∀ i ∈ N. From now on, we assume that R is a Dedekind complete weakly σ-distributive Riesz space.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

RIESZ SPACES

A regulator or (D)-sequence is a bounded double sequence (ai,j)i,j in R s. t. ai,j ↓ 0 ∀ i ∈ N. (D) limn rn = r if ∃ regulator (ai,j)i,j s. t. ∀ ϕ ∈ NN, ∃ n0 ∈ N s. t. |rn − r| ≤

∞

  • i=1

ai,ϕ(i) ∀ n ≥ n0. From now on, we assume that R is a Dedekind complete weakly σ-distributive Riesz space.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

RIESZ SPACES

A regulator or (D)-sequence is a bounded double sequence (ai,j)i,j in R s. t. ai,j ↓ 0 ∀ i ∈ N. (D) limn rn = r if ∃ regulator (ai,j)i,j s. t. ∀ ϕ ∈ NN, ∃ n0 ∈ N s. t. |rn − r| ≤

∞

  • i=1

ai,ϕ(i) ∀ n ≥ n0. R is weakly σ-distributive if for any regulator (ai,j)i,j

  • ϕ∈NN

∞

  • i=1

ai,ϕ(i)

  • = 0.

From now on, we assume that R is a Dedekind complete weakly σ-distributive Riesz space.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

THE FREMLIN LEMMA

Let {(a(p)

i,j )i,j : p ∈ N} be a countable family of

(D)-sequences. Then for each R ∋ u ≥ 0 ∃ regulator (ai,j)i,j s. t., for every ϕ ∈ NN, u ∧

∞

  • p=1

∞

  • i=1

a(p)

i,ϕ(i+p)

  • ≤

∞

  • i=1

ai,ϕ(i).

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

STRUCTURAL ASSUMPTIONS

Fix [a, b] ⊂ R, k ∈ N, and set a ≤ x1,0 < . . . < x1,k ≤ . . . ≤ xn,0 < . . . < xn,k ≤ b, ξi ∈ [xi,0, xi,k], i = 1, . . . , n. The intervals [xi,0, xi,k] form a k-decomposition Π of [a, b], Π := {(ξi; xi,1, . . . , xi,k−1) : [xi,0, xi,k], i = 1, . . . , n}.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

STRUCTURAL ASSUMPTIONS

Fix [a, b] ⊂ R, k ∈ N, and set a ≤ x1,0 < . . . < x1,k ≤ . . . ≤ xn,0 < . . . < xn,k ≤ b, ξi ∈ [xi,0, xi,k], i = 1, . . . , n. The intervals [xi,0, xi,k] form a k-decomposition Π of [a, b], Π := {(ξi; xi,1, . . . , xi,k−1) : [xi,0, xi,k], i = 1, . . . , n}. A k-decomposition is called k-partition if

n

  • i=1

[xi,0, xi,k] = [a, b].

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

STRUCTURAL ASSUMPTIONS

A gauge is a map γ defined in [a, b] and taking values in the set of all open intervals in R, where ξ ∈ γ(ξ) ∀ ξ ∈ [a, b] and γ(ξ) is a bounded open interval for every ξ ∈ R ∩ [a, b].

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

STRUCTURAL ASSUMPTIONS

A gauge is a map γ defined in [a, b] and taking values in the set of all open intervals in R, where ξ ∈ γ(ξ) ∀ ξ ∈ [a, b] and γ(ξ) is a bounded open interval for every ξ ∈ R ∩ [a, b]. Given a gauge γ, a k-decomposition of [a, b] of the type Π = {(ξi; xi,1, . . . , xi,k−1) : [xi,0, xi,k], i = 1, . . . , n} is γ-fine if ξi ∈ [xi,0, xi,k] ⊂ γ(ξi) for all i = 1, . . . , n.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

STRUCTURAL ASSUMPTIONS

A gauge is a map γ defined in [a, b] and taking values in the set of all open intervals in R, where ξ ∈ γ(ξ) ∀ ξ ∈ [a, b] and γ(ξ) is a bounded open interval for every ξ ∈ R ∩ [a, b]. Given a gauge γ, a k-decomposition of [a, b] of the type Π = {(ξi; xi,1, . . . , xi,k−1) : [xi,0, xi,k], i = 1, . . . , n} is γ-fine if ξi ∈ [xi,0, xi,k] ⊂ γ(ξi) for all i = 1, . . . , n. Let [a, b] ⊂ R and δ : [a, b] → R+. A k-partition Π of [a, b] is δ-fine if ξi ∈ [xi,0, xi,k] ⊂ (ξi − δ(ξi), ξi + δ(ξi)) for all i = 1, . . . , n.

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RIESZ SPACES

THE GHk INTEGRAL

Given any k-decomposition of [a, b], Π = {(ξi; xi,1, . . . , xi,k−1) : [xi,0, . . . , xi,k], i = 1, . . . , n} and U : [a, b]k+1 → R, the Riemann sum of U is

  • Π

U :=

n

  • i=1

[U(ξi; xi,1, . . . , xi,k) − U(ξi; xi,0, . . . , xi,k−1)].

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RIESZ SPACES

THE GHk INTEGRAL

A function U : [a, b]k+1 → R is GHk integrable on [a, b] if ∃ I ∈ R and a (D)-sequence (ai,j)i,j such that ∀ ϕ ∈ NN ∃ δ : [c, d] → R+ and P > 0:

  • Π

U − I

  • ≤

∞

  • i=1

ai,ϕ(i) whenever Π is a δ-fine k-partition of any bounded interval [c, d] with [c, d] ⊃ [a, b] ∩ [−P, P]. (GHk) b

a

U := I; U ∈ GHk[a, b]. Analogously we can define the GHk integral for every subinterval of [a, b]. If [a, b] ⊂ R, R = R, k = 1, f : [a, b] → R and U(t, x) = f(t) · x, t, x ∈ [a, b], x = ±∞, then we

  • btain the classical improper Kurzweil-Henstock

integral ([7]).

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

Historical survey

Second Section

RIESZ SPACES

MAIN PROPERTIES

A function U : [a, b]k+1 → R is GHk integrable iff ∃ J ∈ R and a (D)-sequence (ai,j)i,j such ∀ ϕ ∈ NN ∃ a gauge γ s. t.

  • Π

U − J

  • ≤

∞

  • i=1

ai,ϕ(i) whenever Π is a γ-fine partition of [a, b], and in this case b

a

U = J. Cauchy criterion: A map U : [a, b]k+1 → R is GHk integrable iff ∃ (D)-sequence (ai,j)i,j s. t. ∀ ϕ ∈ NN ∃ gauge γ = γ(ϕ) s. t., ∀ γ-fine k-partitions Π, Π′ of [a, b],

  • Π

U −

  • Π′

U

  • ≤

∞

  • i=1

ai,ϕ(i). The GHk integral is a linear monotone functional.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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Second Section

RIESZ SPACES

INTEGRABILITY ON SUBINTERVALS AND ADDITIVITY

If U ∈ GHk[a, b], then U ∈ GHk[c, d] for each [c, d] ⊂ [a, b] w. r. to a same regulator, independent

  • n [c, d].

If U ∈ GHk[a, b] and a < c < b, then (GHk) b

a

U = (GHk) c

a

U + (GHk) b

c

U.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

INTEGRABILITY ON SUBINTERVALS AND ADDITIVITY

Let x0 ∈]a, b[. We say that U satisfies [H1) at x0] if ∃ (D)-sequence (ci,j)i,j (depending in general on x0) s. t. ∀ ϕ ∈ NN ∃ η = η(x0) > 0:

  • [U(x0; w(0)

1 , . . . , w(0) k ) − U(x0; w(0) 0 , . . . , w(0) k−1)]

− [U(x0; w(1)

1 , . . . , w(1) k ) − U(x0; w(1) 0 , . . . , w(1) k−1)]

− [U(x0; w(2)

1 , . . . , w(2) k ) − U(x0; w(2) 0 , . . . , w(2) k−1)]

  • ≤

∞

  • i=1

ci,ϕ(i) whenever

2

  • l=0

k

  • i=1

[w(l)

i−1, w(l) i ]

  • ⊂]x0 − η, x0 + η[ and

w(0) = w(1)

0 , w(0) k

= w(2)

k , x0 = w(1) k

= w(2)

0 .

k = 1: H1) holds anyway; k ≥ 2 and R = R: H1) ⇐ = condition by A. G. Das and S. Kundu.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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INTEGRABILITY ON SUBINTERVALS AND ADDITIVITY

Proposition. Let k ≥ 2, U : [a, b]k+1 → R satisfy H1) at c ∈]a, b[. If U ∈ GHk[a, c] ∩ GHk[c, b], then U ∈ GHk[a, b] and (GHk) b

a

U = (GHk) c

a

U + (GHk) b

c

U.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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THE EXTENSION CAUCHY THEOREM

Let a ∈ R+, U : [a, b]k+1 → R, U ∈ GHk[a, c] ∀ c ∈]a, b[. Assume: H2) ∃ I ∈ R and a (D)-sequence (αi,j)i,j: ∀ ϕ ∈ NN ∃ a left neighbourhood U of b such that

  • (GHk)

c

a

U − I + U(b; y1, . . . , yk−1, b) − U(b; y0, . . . , yk−1)| ≤

∞

  • i=1

αi,ϕ(i) whenever U ∋ c ≤ y0 < y1 < . . . yk−1 < b; H3) ∃ R ∋ u ≥ 0 and gauge γ0 s. t., for every c with a < c < b and every γ0-fine k-partition Π of [a, c],

  • Π

U − (GHk) c

a

U

  • ≤ u.

Then U ∈ GHk[a, b] and (GHk) b

a U = I.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

THE EXTENSION CAUCHY THEOREM

Furthermore, if U ∈ GHk[a, b], then (D) lim

c→b−(GHk)

c

a

U = (GHk) b

a

U (This last assertion holds even if we drop both H2) and H3) ).

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

SOME REMARKS

In several cases in the Riemann sums only the terms with bounded intervals are taken: this means to require that U(±∞; λ1, . . . , λk) = 0 for every choice of λj, j = 1, . . . , k. In this case, H2) can be replaced by: (D) lim

c→+∞(GHk)

c

a

U

  • (D)

lim

d→−∞(GHk)

a

d

U

  • ∃ in R. (1)

R = R and k ≥ 2: H2) ⇐ = (1)∧ condition by A. G. Das & S. Kundu; R = R and k = 1: H2) ⇐ ⇒ condition by ˇ S. Schwabik.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

THE HENSTOCK-SAKS LEMMA

Let U : [a, b]k+1 → R be GHk integrable on [a, b]. Then ∃ (D)-sequence (ai,j)i,j: ∀ ϕ ∈ NN ∃ gauge γ s. t., whenever Π := {(ηi; yi,1, . . . , yi,k−1) : [yi,0, yi,k], i = 1, . . . , m} is a γ-fine k-decomposition of [a, b] (where yi−1,k ≤ yi,0 (i = 2, . . . , m)), then

  • m
  • i=1

[U(ηi; yi,1, . . . , yi,k) − U(ηi; yi,0, . . . , yi,k−1) − (GHk) yi,k

yi,0

U

  • ≤

∞

  • i=1

ai,ϕ(i).

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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MONOTONE CONVERGENCE THEOREM

Let (Un : [a, b]k+1 → R)n. We first introduce the following condition. H4) There exist: U0 : [a, b]k+1 → R; h∗ : [a, b]k+1 → R+; a gauge γ∗

0;

w ∈ R+; a (D)-sequence (a∗

i,j)i,j,

such that:

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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RIESZ SPACES

MONOTONE CONVERGENCE THEOREM

for all γ∗

0-fine k-partitions

Π∗ := {(ti; xi,1, . . . , xi,k−1) : [xi,0, xi,k], i = 1, . . . , q},

q

  • i=1

h∗(ti; xi,1, . . . , xi,k) ≤ w; ∀ ϕ ∈ NN and t ∈ [a, b], ∃ p(t) ∈ N: ∀ n ≥ p(t), for any λ1, . . . , λk, |U0(t; λ1, . . . , λk) − Un(t; λ1, . . . , λk)| ≤ h∗(t; λ1, . . . , λk) ∞

  • i=1

a∗

i,ϕ(i)

  • .
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EXAMPLE

In the case k = 1, H4) is satisfied, when: (Un)n converges to U0 ”w. r. to the same regulator”; and h∗(t, λ) := λ 1 + t2 , t ∈ R; h∗(±∞, λ) = 0, since h(t) := 1 1 + t2 , t ∈ R, is Kurzweil-Henstock integrable on the whole of R, and hence has bounded Riemann sums.

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MONOTONE CONVERGENCE THEOREM

Let (Un : [a, b]k+1 → R)n be a sequence of GHk integrable functions, Un ≤ Un+1, and let

  • (GHk)

b

a

Un

  • n

be bounded. Suppose that ∃ h∗ satisfying H4) together with supn Un =: U0, and H3’) ∃R ∋ w ≥ 0 and gauge γ s. t., for every γ-fine k-partition Π of [a, b],

  • Π

Un − (GHk) b

a

Un

  • ≤ w

∀ n ∈ N. Then U0 is GHk integrable on [a, b] and (GHk) b

a

U0 = (D) lim

n (GHk)

b

a

Un.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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Second Section

RIESZ SPACES

LEBESGUE DOMINATED CONVERGENCE THEOREM

Let (Un : [a, b]k+1 → R)n be a sequence of GHk integrable functions such that

  • n∈P1,m∈P2

|Un − Um| is GHk integrable ∀ P1, P2 ⊂ N; let h∗ : [a, b]k+1 → R+ and supn Un =: U0 satisfy H4). Then U0 is GHk integrable and (GHk) b

a

U0 = (D) lim

n (GHk)

b

a

Un.

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THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction

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References

1 A. BOCCUTO - D. CANDELORO - A. R. SAMBUCINI, Stieltjes-type integrals for metric semigroup-valued functions defined on unbounded intervals, PanAmer. Math. J. (2007), to appear. 2 A. BOCCUTO - B. RIE ˇ CAN - A. R. SAMBUCINI, Some properties of an improper GHk integral in Riesz spaces, Indian J. Math. 50 (2008), to appear. 3 A. G. DAS - S. KUNDU, A generalized Henstock integral, Real Anal. Exch. 29 (2003/2004), 59-78. 4 A. G. DAS - S. KUNDU, A characterization of the GHk integral, Real Anal. Exch. 30 (2004/2005), 639-655. 5 S. PAL - D. K. GANGULY - P. Y. LEE, On convergence for the GR∗

k -integral, Math. Slovaca 55 (2005), 515-527.

6 B. RIE ˇ CAN - T. NEUBRUNN, Integral, Measure and Ordering, Kluwer Academic Publishers/Ister Science, Bratislava, 1997. 7 ˇ

  • S. SCHWABIK, Generalized Ordinary Differential Equations,

World Scientific Publ. Co, Singapore, 1992.