THE GH k INTEGRAL FOR RIESZ RIESZ SPACES SPACE-VALUED FUNCTIONS a - PowerPoint PPT Presentation

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

THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section RIESZ SPACES Introduction

Historical survey THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey REAL-VALUED FUNCTIONS: A. G. DAS, S. Second Section RIESZ SPACES KUNDU (2003-2005);

Historical survey THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey REAL-VALUED FUNCTIONS: A. G. DAS, S. Second Section RIESZ SPACES KUNDU (2003-2005); S. PAL, D. K. GANGULY, P . Y. LEE (2005);

Historical survey THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey REAL-VALUED FUNCTIONS: A. G. DAS, S. Second Section RIESZ SPACES KUNDU (2003-2005); S. PAL, D. K. GANGULY, P . Y. LEE (2005); k = 1: ”generalized Perron integral”, ˇ S. SCHWABIK (1992)

Historical survey THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey REAL-VALUED FUNCTIONS: A. G. DAS, S. Second Section RIESZ SPACES 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).

Historical survey THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey REAL-VALUED FUNCTIONS: A. G. DAS, S. Second Section RIESZ SPACES 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)

Historical survey THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey REAL-VALUED FUNCTIONS: A. G. DAS, S. Second Section RIESZ SPACES 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) GH k -INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS: A. B., B. RIE ˇ CAN, A. R. SAMBUCINI (2007)

Historical survey THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey REAL-VALUED FUNCTIONS: A. G. DAS, S. Second Section RIESZ SPACES 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) GH k -INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS: A. B., B. RIE ˇ CAN, A. R. SAMBUCINI (2007) Extension Cauchy and convergence theorems

Historical survey THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey REAL-VALUED FUNCTIONS: A. G. DAS, S. Second Section RIESZ SPACES 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) GH k -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

RIESZ SPACES THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey A regulator or ( D ) -sequence is a bounded double Second Section sequence ( a i , j ) i , j in R s. t. a i , j ↓ 0 ∀ i ∈ N . RIESZ SPACES From now on, we assume that R is a Dedekind complete weakly σ -distributive Riesz space.

RIESZ SPACES THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey A regulator or ( D ) -sequence is a bounded double Second Section sequence ( a i , j ) i , j in R s. t. a i , j ↓ 0 ∀ i ∈ N . RIESZ SPACES ( D ) lim n r n = r if ∃ regulator ( a i , j ) i , j s. t. ∀ ϕ ∈ NN , ∃ n 0 ∈ N s. t. � ∞ | r n − r | ≤ ∀ n ≥ n 0 . a i ,ϕ ( i ) i = 1 From now on, we assume that R is a Dedekind complete weakly σ -distributive Riesz space.

RIESZ SPACES THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey A regulator or ( D ) -sequence is a bounded double Second Section sequence ( a i , j ) i , j in R s. t. a i , j ↓ 0 ∀ i ∈ N . RIESZ SPACES ( D ) lim n r n = r if ∃ regulator ( a i , j ) i , j s. t. ∀ ϕ ∈ NN , ∃ n 0 ∈ N s. t. � ∞ | r n − r | ≤ ∀ n ≥ n 0 . a i ,ϕ ( i ) i = 1 R is weakly σ -distributive if for any regulator ( a i , j ) i , j � ∞ � � � a i ,ϕ ( i ) = 0 . ϕ ∈ NN i = 1 From now on, we assume that R is a Dedekind complete weakly σ -distributive Riesz space.

THE FREMLIN LEMMA THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section RIESZ SPACES Let { ( a ( p ) i , j ) i , j : p ∈ N } be a countable family of ( D ) -sequences. Then for each R ∋ u ≥ 0 ∃ regulator ( a i , j ) i , j s. t., for every ϕ ∈ NN , � ∞ � � ∞ � � ∞ a ( p ) u ∧ ≤ a i ,ϕ ( i ) . i ,ϕ ( i + p ) p = 1 i = 1 i = 1

STRUCTURAL ASSUMPTIONS THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section Fix [ a , b ] ⊂ � R , k ∈ N , and set RIESZ SPACES a ≤ x 1 , 0 < . . . < x 1 , k ≤ . . . ≤ x n , 0 < . . . < x n , k ≤ b , ξ i ∈ [ x i , 0 , x i , k ] , i = 1 , . . . , n . The intervals [ x i , 0 , x i , k ] form a k -decomposition Π of [ a , b ] , Π := { ( ξ i ; x i , 1 , . . . , x i , k − 1 ) : [ x i , 0 , x i , k ] , i = 1 , . . . , n } .

STRUCTURAL ASSUMPTIONS THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section Fix [ a , b ] ⊂ � R , k ∈ N , and set RIESZ SPACES a ≤ x 1 , 0 < . . . < x 1 , k ≤ . . . ≤ x n , 0 < . . . < x n , k ≤ b , ξ i ∈ [ x i , 0 , x i , k ] , i = 1 , . . . , n . The intervals [ x i , 0 , x i , k ] form a k -decomposition Π of [ a , b ] , Π := { ( ξ i ; x i , 1 , . . . , x i , k − 1 ) : [ x i , 0 , x i , k ] , i = 1 , . . . , n } . A k -decomposition is called k -partition if n � [ x i , 0 , x i , k ] = [ a , b ] . i = 1

STRUCTURAL ASSUMPTIONS THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section A gauge is a map γ defined in [ a , b ] and taking values RIESZ SPACES 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 ] .

STRUCTURAL ASSUMPTIONS THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section A gauge is a map γ defined in [ a , b ] and taking values RIESZ SPACES 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 = { ( ξ i ; x i , 1 , . . . , x i , k − 1 ) : [ x i , 0 , x i , k ] , i = 1 , . . . , n } is γ -fine if ξ i ∈ [ x i , 0 , x i , k ] ⊂ γ ( ξ i ) for all i = 1 , . . . , n .

STRUCTURAL ASSUMPTIONS THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section A gauge is a map γ defined in [ a , b ] and taking values RIESZ SPACES 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 = { ( ξ i ; x i , 1 , . . . , x i , k − 1 ) : [ x i , 0 , x i , k ] , i = 1 , . . . , n } is γ -fine if ξ i ∈ [ x i , 0 , x i , k ] ⊂ γ ( ξ i ) 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 ξ i ∈ [ x i , 0 , x i , k ] ⊂ ( ξ i − δ ( ξ i ) , ξ i + δ ( ξ i )) for all i = 1 , . . . , n .

THE GH k INTEGRAL THE GHk INTEGRAL FOR RIESZ SPACE-VALUED FUNCTIONS Antonio Boccuto Introduction Historical survey Second Section RIESZ SPACES Given any k -decomposition of [ a , b ] , Π = { ( ξ i ; x i , 1 , . . . , x i , k − 1 ) : [ x i , 0 , . . . , x i , k ] , i = 1 , . . . , n } and U : [ a , b ] k + 1 → R , the Riemann sum of U is n � � U := [ U ( ξ i ; x i , 1 , . . . , x i , k ) − U ( ξ i ; x i , 0 , . . . , x i , k − 1 )] . Π i = 1