Measure theory

Measure theory studies functions that assign sizes to sets and the integration and convergence concepts built from those assignments. It provides a rigorous framework for probability and modern integration.

Sigma-algebra: A collection of subsets containing the whole space and closed under complements and countable unions. It specifies which sets a measure is allowed to assign sizes.
Lebesgue measure: The translation-invariant measure on Euclidean space that agrees with ordinary length, area, and volume on suitable sets. It is the standard model for measuring subsets of Euclidean space.
Probability measure: A measure with total mass one, used to assign probabilities to measurable events. Probability theory is measure theory applied to random events.
Riemann integral: An integral defined through limits of finite sums over partitions of an interval. Lebesgue integration handles limits and discontinuities that can defeat Riemann integrability.
Henri Lebesgue: A French mathematician who introduced the integral now bearing his name and helped found modern measure theory. His integration theory made measure a foundation for a broader notion of integral.
Measurable space: A set equipped with a sigma-algebra of distinguished subsets. It is the underlying structure on which measure theory defines measures.
Lebesgue integral: An integral defined by approximating measurable functions with simple functions and using a measure. It extends integration beyond the limits of Riemann integration.
Random variable: A measurable function from a probability space to a measurable space, often the real numbers. Measurability makes probabilities of its outcomes mathematically well-defined.
Riemann–Stieltjes integral: An integral defined by weighting interval partitions according to increments of a function. It is a classical alternative that anticipates integration against measures.
Émile Borel: A French mathematician whose work on sets and functions helped establish the theory of Borel sets. Borel's work supplied a key class of sets later used in measure theory.

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