Radon measureIn 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 topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
Hyperbolic angleIn geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation. The hyperbola xy = 1 is rectangular with a semi-major axis of , analogous to the magnitude of a circular angle corresponding to the area of a circular sector in a circle with radius .
Pushforward measureIn measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function. Given measurable spaces and , a measurable mapping and a measure , the pushforward of is defined to be the measure given by for This definition applies mutatis mutandis for a signed or complex measure. The pushforward measure is also denoted as , , , or .
Hilbert's fifth problemHilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics (for example quark theory) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of group theory and the theory of topological manifolds.
Inner regular measureIn mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. Let (X, T) be a Hausdorff topological space and let Σ be a σ-algebra on X that contains the topology T (so that every open set is a measurable set, and Σ is at least as fine as the Borel σ-algebra on X). Then a measure μ on the measurable space (X, Σ) is called inner regular if, for every set A in Σ, This property is sometimes referred to in words as "approximation from within by compact sets.
Invariant measureIn mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.
Non-measurable setIn mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zermelo–Fraenkel set theory, the axiom of choice entails that non-measurable subsets of exist. The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be measurable.
