In measure theory, a branch of mathematics, Kakutani's theorem is a fundamental result on the equivalence or mutual singularity of countable product measures...

mathematics, Kakutani's theorem may refer to: the Kakutani fixed-point theorem, a fixed-point theorem for set-valued functions; Kakutani's theorem (geometry):...

Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory...

properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated...

Nash equilibria in game theory, and the Arrow–Debreu–McKenzie model of general equilibrium theory in microeconomics. Kakutani's other mathematical contributions...

Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators...

theorem. Angular displacement Complex measure Spectral measure Vector measure Riesz–Markov–Kakutani representation theorem Signed arc length Signed area Signed...

unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in...

measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving...

In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff...

measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory....

Topological combinatorics Necklace splitting problem Ham sandwich theorem Kakutani's theorem (geometry) Imre Bárány Jha, Aditya; Campbell, Douglas; Montelle...

rely on fixed-point theorems such as Brouwer fixed-point theorem for functions (or, more generally, the Kakutani fixed-point theorem for set-valued functions)...

Hadamard's three-circle theorem. On simply connected planar domains, there is a close connection between harmonic measure and the theory of conformal maps....

r>0} , are pre-compact. The Birkhoff–Kakutani theorem (named after mathematicians Garrett Birkhoff and Shizuo Kakutani) states that the following three conditions...

concrete) structure. Cayley's theorem states that every group is isomorphic to a permutation group. Representation theory studies properties of abstract...

founded on the formal definitions of measure theory and dynamical systems, and rather specifically on the notion of a measure-preserving dynamical system. The...

conjecture), the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Bryan Thwaites), Hasse's...

Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among n partners with different valuations in...

Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling...

leaving the measure invariant), and so such systems can have a well-defined entropy independent of the partition. The Ornstein isomorphism theorem states that...

Brouwer's fixed-point theorem into Kakutani's fixed-point theorem. This modification is similar to the generalization of the minimax theorem to the existence...

The Ryll-Nardzewski theorem yields the existence of a Haar measure on compact groups. Fixed-point theorems Fixed-point theorems in infinite-dimensional...

In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes...

implicit function theorem, contraction mappings, measure theory, fixed-point theorems, optimization, and topological degree theory. In particular, equations...

Foundations of the Theory of Probability The theorem has other names including Kolmogorov's consistency theorem, Kolmogorov's extension theorem or the Daniell–Kolmogorov...

an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a...

Probability: Theory and Examples. Cambridge University Press. pp. 191. ISBN 978-1-139-49113-6. Novak, Jonathan (2014). "Pólya's Random Walk Theorem". The American...

mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by Kakutani (1941) and proved...

In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties...