For non-additive set functions, the independent product, in general, is not unique and the Fubini theorem is restricted to slice-comonotonic functions. In this paper, we use the representation theorem of Gilboa and Schmeidler (Math Oper Res 20:197–212, 1995) to extend the Möbius product for non-additive set functions to non-finite spaces. We extend the uniqueness result of Ghirardato (J Econ Theory 73:261–291, 1997) for products of two belief functions and weaken the requirements on the marginals necessary to obtain (...) the Fubini property in the product. More importantly, we show that for the Möbius product one side of the Fubini theorem holds for all integrable functions if one of the marginals either is a probability or a convex combination of a chain of unanimity games, i.e., we relax the requirement of slice-comonotonicity and enrich the set of possible applications. (shrink)

We study the notion of [Formula: see text]-MAD families where [Formula: see text] is a Borel ideal on [Formula: see text]. We show that if [Formula: see text] is any finite or countably iterated Fubini product of the ideal of finite sets [Formula: see text], then there are no analytic infinite [Formula: see text]-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective [Formula: see text]-MAD families; and under the full Axiom of Determinacy [Formula: (...) see text][Formula: see text] or under [Formula: see text] there are no infinite [Formula: see text]-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal [Formula: see text], which corresponds to the classical notion of MAD families, as well as to the ideal [Formula: see text]. The proofs combine ideas from invariant descriptive set theory and forcing. (shrink)

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices (...) of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails. (shrink)

We present several naturally defined σ-ideals which have Borel bases but, unlike for the classical examples, these ideals are not of bounded Borel complexity. We investigate set-theoretic properties of such σ-ideals.

We investigate higher dimensional chain conditions, where the largeness notion is given by Fubini products of a given ideal. From strong saturation properties of an ideal, we derive abstractly versions of higher dimensional [Formula: see text]-system lemma, which imply many posets, including any finite support iteration of [Formula: see text]-centered posets and measure algebras, satisfy the higher dimensional chain conditions. We then show that if a poset satisfies a strengthening of the [Formula: see text]-finite chain condition by Horn and (...) Tarski, then it satisfies higher dimensional chain conditions. As an application, we derive Ramsey-theoretic consequences, namely various partition hypotheses as studied by Bannister, Bergfalk, Moore and Todorcevic, from the existence of ideals satisfying strong chain conditions. (shrink)

Journal of Mathematical Logic, Ahead of Print. We investigate higher dimensional chain conditions, where the largeness notion is given by Fubini products of a given ideal. From strong saturation properties of an ideal, we derive abstractly versions of higher dimensional [math]-system lemma, which imply many posets, including any finite support iteration of [math]-centered posets and measure algebras, satisfy the higher dimensional chain conditions. We then show that if a poset satisfies a strengthening of the [math]-finite chain condition by Horn (...) and Tarski, then it satisfies higher dimensional chain conditions. As an application, we derive Ramsey-theoretic consequences, namely various partition hypotheses as studied by Bannister, Bergfalk, Moore and Todorcevic, from the existence of ideals satisfying strong chain conditions. (shrink)

A \(\sigma \) -ideal \(\mathcal {I}\) on a Polish group \((X,+)\) has the Smital Property if for every dense set _D_ and a Borel \(\mathcal {I}\) -positive set _B_ the algebraic sum \(D+B\) is a complement of a set from \(\mathcal {I}\). We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are \(\mathfrak {c}\) many maximal invariant (...) \(\sigma \) -ideals with Borel bases on the Cantor space \(2^\omega \). (shrink)

We study ultrafilters onω2produced by forcing with the quotient of${\cal P}$(ω2) by the Fubini square of the Fréchet filter onω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated (...) but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters. (shrink)

In questo volume Enrico Fubini si propone di mettere in luce i temi fondamentali, le correnti piú importanti, i problemi chiave dell'estetica musicale contemporanea, iniziando il loro studio dall'illuminismo, in cui tali problemi affondano le loro radici. Il concetto di estetica è inteso nel senso piú largo, comprendendo le meditazioni sulla musica non solo degli studiosi di estetica e dei musicisti stessi, ma anche dei filosofi, degli psicologi, dei fisici e dei matematici.

[ES] Este trabajo parte de la pregunta sobre si el arte musical es capaz de expresar realmente el sentimiento trágico, e intenta clarificar la relación que existe entre el concepto de lo trágico y el lenguaje de la música. En concreto, se trata de reconstruir la historia de esta relación empezando por sus raíces, hundidas en la tragedia griega, para llegar a sus más cercanas expresiones en el siglo XX. Se pone el acento en el valor trágico del propio contenido (...) musical, destacando con atención, en el caso del melodrama, la relación que surge entre una emoción expresada en palabras y aquella que nace de la música misma, así como la contradicción aparente entre la belleza de la armonía de la música polifónica con el sentimiento de lo verdaderamente trágico. [EN] This writing raises the question if the musical art is able to really express the feeling of the tragic, and tries to clarify the relation that exist between the concept of the tragic and the language of music. In particular, it aims to reconstruct the history of this relation beginning by its roots, sunk in Greek tragedy, to arrive at its nearer expressions in the XX century. The accent is on the tragic value of the musical content itself, emphasizing in particular, in the case of the melodrama, the relation that arise between an emotion expressed in words and that that is born of music itself, as well as the apparent contradiction between the beauty of the harmony of polyphonic music and the feeling of the truly tragic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Humanism and Truth: Valla Writes against the Donation of ConstantineRiccardo FubiniTranslated by Anastasia Ananson and William ConnellThere has existed for a long time now in studies of Renaissance humanism (and not only as these have developed in a single country or disciplinary area) a tendency to consider from a prevalently formalist point of view what was instead an innovative and complex cultural experience. A particularly privileged position has been (...) reserved for the ars rhetorica, which, considered in its “purely formal instrumentality,” has been treated as though immune from ideological contamination. 1 The fear of becoming entangled in matters of ideological interpretation may, in turn, give rise to new and more seriously flawed readings. Although this is a matter which I shall not now emphasize, it should be clear that when the interpretation of a text is limited to its “purely formal instrumentality,” the vacuum that results can be filled more easily with matter that is new and different. Ideology chased from the doorstep may yet sneak in through the window, sometimes completely stripped of its real historical context. However, what is more important to note here as a premise to my discussion is that, along with the formalism already referred to, one finds in recent studies a tendency to diminish or minimize the significance of writings that were celebrated by the historical traditions of the last century and felt to belong to a shared cultural, religious, or political legacy.Among such writings the De falso credita et ementita Constantini donatione of Lorenzo Valla has certainly played an important part, beginning at least with the clamor that was raised on its behalf by Ulrich von Hutten and Martin Luther. Yet it is quite typical that the work’s most recent [End Page 79] editor, who has dedicated much careful research to Valla’s treatise and its reception, should have attempted to disconnect the meaning of Valla’s treatise in its own time from the influence it would later have, concluding that “the influence [of Valla’s work] manifested itself in accents quite different from those that have resulted from [current] interpretation.” 2 In the view of this scholar, at the time of the composition of Valla’s work, Constantine’s alleged “donation” was no longer a matter of contemporary relevance; rather, it furnished the theme for a brilliant exercise in legal rhetoric. He writes: “By then the Donation of Constantine was no longer an urgent theme for political theory. (Die Konstantinische Schenkung war damals kein beherrschendes Thema der politischen Theorie mehr.)” 3My discussion takes its cue from this point. Without dwelling at length on the various contributions of other scholars, I shall begin first by trying, on the basis of documentary evidence, to restore Valla’s treatise to life, to the climate of its day, in order then to gather and present its meaning along interpretative lines I have already laid out elsewhere. 4According to that usually trustworthy Florentine chronicler, Giovanni Cavalcanti, Filippo Maria Visconti, duke of Milan, at the end of 1443 made a diplomatic avance toward Florence. Principal among the issues dividing the two powers had been the protection afforded by the regime of Cosimo de’ Medici to Francesco Sforza, then holed up in his fiefdoms in the March of Ancona. The Duke now proposed that they should unite in common defense against the Pope, insofar as the latter was sovereign lord of the March. Beyond these particular circumstances, the accord was supposed to find its warrant in the partners’ shared interest in affirming the superiority of civil over ecclesiastical jurisdiction. So the Duke, according to Cavalcanti, advanced his proposal with words that allow us to suspect an early but not surprising echo of Valla’s treatise (first circulated in that very year): “It so happens that even if Constantine consigned to Sylvester so many and such rich gifts—which is doubtful, because such a privilege can nowhere be found—he could only have granted them for his [Sylvester’s] lifetime, [because] the Empire takes precedence over any lordship.” No differently from Valla, Duke Filippo Maria thought the sovereign impersonality of the [End Page 80] law was the prerogative only of the Empire (and... (shrink)

The Renaissance movement known as humanism eventually spread from Italy through all of western Europe, transforming early modern culture in ways that are still being felt and debated. Central to these debates—and to this book—is the question of whether the humanist movement contributed to the secularization of Western cultural traditions at the end of the Middle Ages. A preeminent scholar of Italian humanism, Riccardo Fubini approaches this question in a new way—by redefining the problem of secularization more carefully to (...) show how humanists can at once be secularizers and religious thinkers. The result is a provocative vision of the humanist movement. _Humanism and Secularization_ offers a nuanced account of humanists contesting medieval ideas about authority not in order to reject Christianity or even orthodoxy, but to claim for themselves the right to define what it meant to be a Christian. Fubini analyzes key texts by major humanists—isuch as Petrarch, Poggio, and Valla—from the first century of the movement. As he subtly works out these authors’ views on religion and the Church from both biographical and textual information, Fubini reveals in detail the new historical consciousness that animated the humanists in their reading of classical and patristic texts. His book as a whole shows convincingly just how radical the humanism of the first half of the fifteenth century was and how sharply it challenged well-entrenched ideas and institutions. Appearing here in English for the first time, his work provides a model set of readings of humanist texts and a critical perspective on Italian humanism that will alter and enrich discussion and understanding of the nature of the humanist movement. (shrink)

Il volume affronta i complessi problemi sollevati dalla musica a studiosi, critici, filosofi, e che riguardano cio che ruota attorno a questa arte dalle molteplici facce. Punto chiave della riflessione sono le avanguardie del Novecento, poiché le sue profonde trasformazioni mettono a nudo ed evidenziano, a volte in modo drammatico, gli interrogativi di fondo a cui la musica ci ha sempre posto di fronte. Il tema centrale è il rapporto tra una supposta universalità dell'esperienza musicale in senso lato e la (...) sua innegabile storicità, tema su cui si dibatte nel pensiero occidentale sin dai tempi di Pitagora. Pur essendo un linguaggio chiuso nella sua specificita, la musica ci spinge di continuo ad andare oltre: allude sempre a qualcosa che sta al di fuori dei suoni stessi e ci invita a esplorare i luoghi a cui segretamente tende"--Back cover. (shrink)

Research of the complex relations which there are between “theory of affections/effects in music” and “theory of harmony of the world” . Both are approaches which go back to Renaissance and cover Baroque and Enlightenment until arriving to Romanticism. It is an efficacious meeting between an empiricist trend and a mathematical rational trend in music.

A history of music aesthetics from Antiquity to the present day. Fubini considers musical thought in terms of its relationship to philosophy, medicine, astronomy, mathematics and teaching in the period up to the Enlightenment. The second half of the book concentrates on key issues in contemporary music aesthetics, placing them in the perspective of the Enlightenment and of 19th-century philosophers.".

In the history of the Hebrew Thought there are some points about the value of music and chant wich can seem contradictory. Generally speaking, the rationalistic thought tends to forbid music, especially in profane uses because of its lascivious character. On the contrary, mystical thought tends to considerer music and chant as a way of spiritual rising. The Zohar, the great book of the Hebrew medieval mysticism, considers music as an esoteric thing related wich mysticism and prophecy.

What concept of Judaism is present in Schönberg’s philosophy of music? It is impossible to separate the musical texture from his experience of reconciliation with Judaism, and his new idea of musical drama is a confirmation that the dodecaphonic structure of musical thinking connects with Schönberg’s idea of the Jewish ethical and religious point of view. A comparative analysis of some essays with some operas shows the internal tie between music and Judaism in dodecaphony.