In this talk, we discuss an abstract approach to entropy in countable measurable and induced probability spaces. We consider applications and interpretations of this approach in the context of Rough Set Theory, Fuzzy Set Algebras, as well as Conservative Classical Logics and Quantum Computing. The talk consists of three parts. The first part considers entropy as associated with measure distributions understood as sequences of non-negative values. The second part does the same for partitions. Finally, the third part refers to the aforementioned applications.

Cattaneo’s scientific life can be divided into two periods of different activity: (1) 1973-1983, Associate Professor of Mathematical Methods of Physics at the University of Calabria, with scientific researches on "Axiomatic Foundations of Quantum Mechanics". (2) 1984-1988, Associate Professor and from 1999 until his retirement in the year 2013, Full Professor of Discrete Time Dynamical Systems (applications to neural networks and Chaos) at the University of Milano-Bicocca. During his stay at the University of Milano-Bicocca, he founded and directed the "FisLab-Laboratory: Parallel Computational Physics of Complex Systems", financed by various Italian industries.

In 1990 in Gdansk (Poland), during an international congress, he was one of the promoters and founder of the "International Quantum Structures Association (IQSA)" of which he was a member of the Scientific Committee from 1990 until 2004.

In addition to other official positions, he was a member of the "Editorial Board of the Transactions on Rough Sets" (LNCS). Head of the "International Federation of Information Processing (IFIP)" Working Group on the topic Cellular Automata and Discrete Complex Systems. Local Head (University of Milano-Bicocca) of the International Advisory Board-European School of Advanced Studies in "Methods for Management of Complex Systems", University of Pavia and Milano-Bicocca.

Responsible of Scientific Projects. Cattaneo has been responsible for many scientific projects of national and international importance, of which we mention the most important.

Research Projects of National Interest (PRIN) on Mathematical Problems of Quantum Mechanics (1978-1984) and on Computational Models in the Simulation of Complex Dynamic Systems (1984-2013). Local responsible of the National Research Council (CNR) project on Computer Systems and Parallel Computation (1988-1944). From 1991 to 1993 { External responsible for a project involving the DBit (Department of Biotechnology) of the San Raffaele hospital in Milan, the Joint Disease Hospital (New York University), and the IBM Watson Research Center (Theoretical Physics Department of New York), concerning the study by Artificial Intelligence methods of the Immune system under the invasion of antigens.

From 1991 to 1997 – Responsible for a research contract with the "Company Illycaffe" S.p.A. (Trieste), the "Consortium for Automatic Calculation" (CINECA - Bologna) and the "CRAY Supercomputer Research Center" (Bologna), on Simulations of percolation processes in coffee porous media by lattice gas techniques with cellular automata. From 1993 to 1995 – Collaborator of the English project: "Modeling in Economics and Physics", at the London School of Economics (LSE). From 1999 to 2000 – Responsible of the international research contract with "ELF Petrole Aquitaine" S.E. (France), on the Potential evaluation of the new representation methods of the dynamics of gas and liquid dual flow in the instillation or absorption columns.

From 2003 to 2007 he was responsible of the Milano-Bicocca COFIN project - "Mathematical and Computer Sciences" entitled Automata and Formal Languages: mathematical and applicative aspects on the following topics: (1) Algebraic approach to fuzzy logic; (2) Rough set theory with applications; (3) Reversible logic and computational conservativeness – towards Quantum Computing; (4) Cellular Automata – chaotic symbolic dynamics and associated languages. In 1984, for the first time, Cattaneo introduced the Brouwer-Zadeh lattice structure as an algebra having fuzzy sets as a model, and rough sets as application domain.

Conclusions on the research activity. Cattaneo has carried out an intense research activity, with results published in international journals of about 210 scientific publications, with more than 2,700 citations, about 18,000 readers, an H index of 27 and a Research Interest Score of 1,245 (higher than 94% of members).