Measurable function

A function between measurable spaces is measurable when the inverse image of every measurable set in its codomain is measurable in its domain.

Sigma-algebra: A collection of subsets containing the whole space and closed under complements and countable unions. It specifies which domain and codomain subsets count as measurable.
Preimage characterization of continuity: A topological characterization stating that a function is continuous when inverse images of open sets are open. It parallels measurability while replacing open sets with measurable sets.
Lebesgue integral: An integral that extends integration through measure and approximation by simple functions. Measurability is a central prerequisite for defining this integral.
Continuous function: A function whose inverse images of open sets are open. Continuity uses topology, whereas measurability uses specified sigma-algebras.
Measurable space: A set equipped with a sigma-algebra of designated measurable subsets. A measurable function is defined between spaces equipped with these structures.
Composition of measurable functions: The composition of measurable maps is measurable when their measurable-space structures match. Nested inverse images show why measurability survives composition.
Random variable: A measurable function from a probability space to a measurable value space. Random variables are measurable functions that encode outcomes numerically or categorically.
Borel measurable function: A function whose inverse images of Borel sets are measurable in its domain. This common special case fixes the codomain sigma-algebra to its Borel sets.
Inverse image: For a function and a subset of its codomain, the inverse image contains all domain points mapped into that subset. The defining condition tests inverse images, not direct images.
Almost-everywhere convergence: A sequence of functions converges at every point outside a set of measure zero. Pointwise limits of measurable functions remain measurable, supporting this convergence framework.

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