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The integral of a measurable function with respect to a measure, built from simple functions and limits; for signed or complex functions it is defined when the relevant positive and negative parts are integrable.
tao-measure-theoryEach relation below ends at this concept.
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Lebesgue measure supplies the standard volume measure used in the Lebesgue integral on Euclidean space; other measures give different, equally valid integrals.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
tao-measure-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Measurability is the structural condition that permits approximation by simple functions and defines the Lebesgue integral when the relevant extended-real integrals are finite or well specified.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
tao-measure-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
A measure space supplies the measure against which measurable functions are integrated; changing that measure generally changes both integrability and the resulting value.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
tao-measure-theoryEach relation below starts at this concept.
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