measure theory

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measure theory

mathematics

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June 30, 2026

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Measure theory is a branch of mathematics that provides a rigorous foundation for analysis and probability. It generalizes the concepts of length and area to arbitrary sets of points, allowing for the measurement of complex sets that are not easily defined by simple intervals or rectangles.12

Developed significantly by Henri-Léon Lebesgue, measure theory addresses limitations in earlier definitions of the integral, such as the Riemann integral. It allows for the integration of functions that may oscillate wildly, by focusing on the "size" or "measure" of sets of points. For instance, the set of rational numbers, though infinite, has a measure of zero, meaning it is considered negligible in certain mathematical contexts.132

This theory is fundamental to modern probability theory, providing a framework for defining probabilities on infinite sample spaces. It was notably used by Andrey Kolmogorov to establish probability theory on a rigorous mathematical footing. Measure theory also finds applications in areas like Brownian motion and quantum electrodynamics.43

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analysis
Measure theory
A rigorous basis for the new discipline of analysis was achieved in the 19th century, in particular by the German mathematician Karl Weierstrass. Modern analysis, however, differs from that of Weierstrass’s time in many ways, and the most obvious is the level of…
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A rigorous basis for the new discipline of analysis was achieved in the 19th century, in particular by the German mathematician Karl Weierstrass. Modern analysis, however, differs from that of Weierstrass’s time in many ways, and the most obvious is the level of…
Read More

Achilles paradox
In an anticipation of modern measure theory, Aristotle argued that an infinity of subdivisions of a distance that is finite does not preclude the possibility of traversing that distance, since the subdivisions do not have actual existence unless something is done to them, in this case stopping at them. See…
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In an anticipation of modern measure theory, Aristotle argued that an infinity of subdivisions of a distance that is finite does not preclude the possibility of traversing that distance, since the subdivisions do not have actual existence unless something is done to them, in this case stopping at them. See…
Read More

probability theory
Measure theory
During the two decades following 1909, measure theory was used in many concrete problems of probability theory, notably in the American mathematician Norbert Wiener’s treatment (1923) of the mathematical theory of Brownian motion, but the notion that all problems of probability theory could…
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During the two decades following 1909, measure theory was used in many concrete problems of probability theory, notably in the American mathematician Norbert Wiener’s treatment (1923) of the mathematical theory of Brownian motion, but the notion that all problems of probability theory could…
Read More

mathematics
Riemann’s influence
In this theory, called measure theory, there are sets that can be measured, and they either have positive measure or are negligible (they have zero measure), and there are sets that cannot be measured at all.
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In this theory, called measure theory, there are sets that can be measured, and they either have positive measure or are negligible (they have zero measure), and there are sets that cannot be measured at all.
Read More