406 research outputs found
MR2741185 Talvila, Erik The regulated primitive integral. Illinois J. Math. 53 (2009), no. 4, 1187–1219. (Reviewer: Luisa Di Piazza) 46G12 (26A39 46E15 46F10)
No full textTalvila Erik, The regulated primitive integral. Illinois J. Math. 53 (2009), no. 4, 1187–1219, 46Gxx (26A39 46Exx) MR 2 741 185
A descriptive definition of an integral is a definition which provides a ``description'' of the space of primitives. The derivatives in some sense of the primitives are the integrands.
In this paper the author introduces a descriptive method of integrating distributions: the regulated primitive integral.
The set \textbf{B}_R= \{F: [-\infty,\infty]
\rightarrow {\bf R}
\ \ | \mbox{ F {\it is regulated and left continuous on }}\\
\ \ {\bf R},
\ \ F(-\infty)=0, \ \ F(\infty)\in {\bf R}\} is the family of primitives.
The derivative here is in the sense of the distributions (i.e. a distributional or weak derivative). Then the integrable distributions are those distributions (in the Schwartz's sense) that are the distributional derivative of a function in .
The regulated primitive integral is a proper extension of the integral of distribution defined by L. Schwartz [Théorie des distributions. (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X. Hermann, Paris 1966 xiii+420, 46.40 (44.00), MR0209834 (35730)]. Moreover it is proved that the space of regulated integrable distributions is the completion of the space of signed Radon measures in the Alexiewicz norm, but it is not the completion in this norm of the Henstock-Kurzweil integrable functions. The functions of bounded variation constitute its dual space and also the space of multipliers. In the introduction a wide
panorama of descriptive and constructive integration methods is given.
Reviewed by (L. Di Piazza
MR2569913: Rodríguez, José. Some examples in vector integration. Bull. Aust. Math. Soc. 80 (2009), no. 3, 384–392. (Reviewer: Luisa Di Piazza),
No full textThe paper deals with some classical examples in vector integration due to Phillips, Hagler and Talagrand, revisited from the point of view of the Birkhoff and McShane integrals. More precisely, the author considers:
- Phillips' example of a Pettis integrable function f which is not Birkhoff integrable [R. S. Phillips, Trans. Amer. Math. Soc. 47 (1940), 114--145; MR0002707 (2,103c)]. It is proved here that f is universally McShane integrable.
- Hagler's example of a scalarly measurable l∞-valued function g which is not strongly measurable. The function g is proved to be universally Birkhoff integrable.
- Talagrand's example of a bounded Pettis integrable function φ having no conditional expectation [M. Talagrand, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224 pp.; MR0756174 (86j:46042)]. Here the author shows that φ is also Birkhoff integrable, giving a negative answer to the question whether conditional expectations exist within the Birkhoff theory.
Some interesting open problems are also stated.
Reviewed by Luisa Di Piazz
MR2553995 (2010h:26008): Mihail, Alexandru The Arzela-Ascoli theorem for partial defined functions. An. Univ. Bucureşti Mat. 57 (2008), no. 2, 259–268. (Reviewer: Luisa Di Piazza),
No full textIn this paper the author gives a generalization of the Arzela-Ascoli theorem for partial defined functions, i.e., for functions defined in a nonempty subset of a metric space X and taking values in a metric space Y. To this end suitable definitions of local and uniform convergence for partial defined functions are introduced. As an application a different proof of a known result concerning the existence of Lipschitz selections for Lipschitz multifunctions is given.
Reviewed by Luisa Di Piazz
MR2886259 Naralenkov, Kirill Several comments on the Henstock-Kurzweil and McShane integrals of vector-valued functions. Czechoslovak Math. J. 61(136) (2011), no. 4, 1091–1106. (Reviewer: Luisa Di Piazza) 26A39 (28B05)
No full textIn this paper the author essentially discusses the difference between the Henstock-Kurzweil and McShane integrals of vector-valued functions from the descriptive point of view. He first considers three notions of absolute continuity for vector-valued functions AC, AC*, AC_{\delta}) and studies the relationships between the corresponding classes of functions. Then he uses such notions to give descriptive characterizations of the Henstock-Kurzweil and McShane integrable functions
MR3191427 Naralenkov, Kirill M., A Lusin type measurability property for vector- valued functions. J. Math. Anal. Appl. 417 (2014), no. 1, 293307. 28A20
No full textIn the paper under review the author introduces the notion of Riemann measurability for vector-valued
functions, generalizing the classical Lusin condition, which is equivalent to the Lebesgue measurability
for real valued functions. Let X be a Banach space, let f : [a; b] ! X and let E be a measurable subset of [a; b]. The function
f is said to be Riemann measurable on E if for each " > 0 there exist a closed set F E with
(E n F) < 0 (where is the Lebesgue measure) and a positive number such that
k XK
k=1
ff(tk) ?? f(t0
k)g (Ik)k < "
whenever fIkgKk
=1 is a nite collection of pairwise non-overlapping intervals with max1 k K (Ik) <
and tk; t0
k 2 Ik
T
F.
The Riemann measurability is more relevant to Riemann type integration theory, such as those of
McShane and Henstock, rather than the classical notion of Bochner or scalar measurability. In par-
ticular the author studies the relationship between the Riemann measurability and the M and the H
integrals that are obtained if we assume that the gauge in the de nitions of McShane and Henstock
integral can be chosen Lebesgue measurable.
The main results are the following
If f : [a; b] ! X is H-integrable on a measurable subset E of [a; b], then f is Riemann measurable
on E.
If f : [a; b] ! X is both bounded and Riemann measurable on a measurable subset E of [a; b], then
f is M-integrable on E.
If f : [a; b] ! X is both Riemann measurable and McShane (Henstock) integrable on a measurable
subset E of [a; b], then f is M-integrable (H-integrable) on E.
Suppose X separable. If f : [a; b] ! X is McShane (Henstock) integrable, then f is M-integrable
(H-integrable.)
The author concludes the paper with the following open problem: for which families of non-separable
Banach spaces does the McShane (or even the Pettis) integrability imply Riemann measurability?
Reviewed by (L. Di Piazza
MR2684422 Deville, Robert; Rodríguez, José Integration in Hilbert generated Banach spaces. Israel J. Math. 177 (2010), 285–306. (Reviewer: Luisa Di Piazza)
No full text2010), 285–306, 46Exx (46J10)
It is known that each McShane integrable function is also Pettis integrable, while the reverse implication in general is not true.
The equivalence of McShane and Pettis integrability depends on
the target Banach space X and has been proven: by R. A. Gordon [Illinois J. Math. 34 (1990), no. 3, 557–567, 26A42 (28B15 46G10 49Q15)], and by D. H. Fremlin and J. Mendoza [Illinois J. Math. 38 (1994), no. 1, 127–147, 46G10 (28B05)] if X is separable, by D. Preiss and the reviewer [Illinois J. Math. 47 (2003), no. 4, 1177–1187. 28B05 (26A39 26E25 46G10)] if X=c_0(\Gamma) (for any set
\Gamma) or X is super-reflexive, by the second author of the present paper [J. Math. Anal. Appl. 341 (2008), no. 1, 80–90, 46G10 (28B05 46B99 47B10)] if X=L^1(\nu) (for any probability measure
\nu).
Here the authors show that the McShane and Pettis integrability coincide for functions taking values in a subspace of a Hilbert generated Banach space. This result includes all previous known ones concerning the above mentioned equivalence.
The used approach relies heavily on some special properties of the Markushevich bases of those Banach spaces. They also give a ZFC example of a scalarly negligible function which is not McShane integrable.
Moreover they prove that, whenever the target Banach space is super-reflexive generated, the Birkhoff integrability lies strictly between Bochner and McShane integrability. Reviewed by L. Di Piazz
MR2481817 (2010e:46040): Haluška, Ján; Hutník, Ondrej On vector integral inequalities. Mediterr. J. Math. 6 (2009), no. 1, 105–124. (Reviewer: Luisa Di Piazza),
No full textI. Dobrakov in his papers [Czechoslovak Math. J. 40(115) (1990), no. 1, 8--24; MR1032359 (90k:46097); Czechoslovak Math. J. 40(115) (1990), no. 3, 424--440; MR1065022 (91g:46052)] developed a theory for integrating vector-valued functions with respect to operator-valued measures:
Let X and Y be two Banach spaces, Δ be a δ-ring of subsets of a nonempty set T, L(X,Y) be the space of all continuous operators L:X→Y, and m:Δ→L(X,Y) be an operator-valued measure σ-additive in the strong operator topology of L(X,Y). A measurable function f:T→X is said to be integrable in the sense of Dobrakov if there exists a sequence of simple functions fn:T→X, n∈N, converging m-a.e. to f and the integrals ∫.fndm are uniformly σ-additive measures on σ(Δ) (i.e. the σ-algebra generated by Δ). The integral of the function f on E∈σ(Δ) is defined by the equality ∫Efdm=limn→∞∫Efndm.
The first author, in the papers [Math. Slovaca 43 (1993), no. 2, 185--192; MR1274601 (95f:46069); Rev. Roumaine Math. Pures Appl. 38 (1993), no. 4, 327--337; MR1258045 (95g:28008); Czechoslovak Math. J. 47(122) (1997), no. 2, 205--219; MR1452416 (98h:46044)], has generalized the Dobrakov integration to the complete bornological locally convex topological vector spaces (C.B.L.C.S., for short), developing a new technique for C.B.L.C.S. and operator-valued measures. The main novelty of such a theory is that instead of the "classical'' objects such as a submeasure, a norm, a metric, etc., he needs to work with lattices of submeasures, norms, etc.
In the present paper the authors first give a brief development of this new integration theory in C.B.L.C.S. (see Section 2). In Section 3 some inequalities, which are important tools in this Dobrakov-type integration technique, are proved.
Reviewed by Luisa Di Piazz
In dialogo su libertà di espressione, riso e violenza
No full textIl saggio si inserisce in un volume in cui dialogano filosofi del diritto e filosofi del linguaggio. Nello specifico lo scritto di Clelia Bartoli discute con quello di Salvatore Di Piazza il cui cuore tematico riguarda le dinamiche che si generano all’interno di un triangolo i cui vertici sono costituiti da tre concetti: il riso, la libertà di espressione e la violenza.
L’argomentazione del filosofo del linguaggio usa come banco di prova un caso emblematico: «La satira à la Charlie Hebdo, ovvero la presa in giro irriverente, caustica e politicamente scorretta». Di Piazza debutta con un paio di domande: «si può ridere di tutto?» e «si può dire di tutto in un discorso connesso al riso?». In altri termini, l’autore domanda entro quali confini si può muovere la libertà di espressione che utilizzi il registro comico affinché non sia lesiva della libertà e della dignità altrui.
Clelia Bartoli mostra come l’autore parta da una concezione liberale della libertà di espressione che dalla Rivoluzione francese in poi è la concezione dominante nella produzione normativa. Ma propone di affrontare il tema con una concezione della libertà di espressione differente, che non abbia a modello la pars condicio di matrice liberale, bensì il dialogo autentico che emerge dalla proposta ermeneutica di Gadamer. Da ciò deriverebbero implicazioni normative alternative per definire la libertà di espressione.The essay is part of a volume in which philosophers of law and philosophers of language dialogue. Specifically, Clelia Bartoli's paper addresses Salvatore Di Piazza’s writing whose thematic core concerns the dynamics generated within a triangle whose vertexes consist of three concepts: laughter, freedom of expression and violence.
The philosopher of language's argument uses as a crucial case study: "Satire à la Charlie Hebdo, or the irreverent, caustic and politically incorrect mockery". Di Piazza debuts with a couple of questions: "can one laugh at everything?" and "can one say everything in a discourse related to humor?". In other words, the author asks within which boundaries freedom of expression using the comic register can be moved so that it is not detrimental to the freedom and dignity of others.
Clelia Bartoli shows how the author starts from a liberal conception of freedom of expression, which from the French Revolution onwards is the conception that inspires most of the normative production. But she proposes to approach the issue with a different conception of freedom of expression, one that does not model itself on liberal idea of the “pars condicio”, but on the “authentic dialogue” as it emerges from Gadamer's hermeneutic proposal. This would give rise to alternative normative implications for defining freedom of expression
Camera oscura e interiorità
No full textThe essay thematizes the complex relationship between science, literature and philosophy will go around the question of the representation of interiority. In these pages, the author shows the novelty of an interdisciplinarity approach capable of grasping the contributions of scientifc discourse and the literary paradigm in the construction of interiority form the end of the Eighteenth Century. At hte same time the author outlines the conceptual and historical scenario in which situate the various contributions that compose this review's issue respect to which the essay is also the introduction
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