THE SEMIGROUP OF METRIC MEASURE SPACES AND ITS INFINITELY DIVISIBLE PROBABILITY MEASURES

Area of Science:

• Probability theory
• Measure theory
• Metric geometry

Background:

• Introduces metric measure spaces as complete, separable metric spaces with a full-support probability measure.
• Defines equivalence of such spaces via measure-preserving isometries.
• Highlights the Gromov-Prohorov metric for the space of equivalence classes.

Purpose of the Study:

• To investigate the algebraic and analytical properties of metric measure spaces under a novel Cartesian product operation.
• To explore the structure of the resulting Polish semigroup, including factorization and divisibility properties.
• To analyze the interaction between the semigroup structure and metric scaling, and to study probability measures and processes on this space.

Main Methods:

• Defined a binary operation (Cartesian product) on metric measure spaces, equipping it with the sum of metrics and product of probability measures.
• Utilized continuous semicharacters to analyze semigroup properties, including unique factorization and absence of infinitely divisible elements.
• Investigated the action of positive real numbers (metric scaling) on the semigroup and characterized specific probability measures and processes.

Main Results:

• Established that metric measure spaces with the defined operation form a cancellative, commutative, Polish semigroup with a translation-invariant metric.
• Demonstrated the absence of infinitely divisible elements and the existence of unique prime factorizations.
• Showed that the law of large numbers does not hold for sequences of random spaces, and characterized infinitely divisible, Lévy, and stable probability measures.

Conclusions:

• The study introduces a rich algebraic structure (Polish semigroup) on metric measure spaces, enabling deep analytical investigations.
• The unique factorization and lack of infinitely divisible elements highlight the distinct nature of this mathematical object.
• Characterizations of probability measures and processes provide foundational results for further research in this area.