Almost everywhere

A property holds almost everywhere on a measure space when the set of points where it fails has measure zero. It allows exceptions that are negligible under the chosen measure.

Measure space: A set equipped with a collection of measurable subsets and a measure assigning them sizes. Almost-everywhere claims are defined relative to the measure space in which exceptions are measured.
Almost-sure convergence: Convergence of random variables at every outcome except possibly on an event of probability zero. It is pointwise convergence expressed as an almost-everywhere claim on a probability space.
Lebesgue integral: An integral defined by measuring level sets of measurable functions, including functions with broad or irregular domains of support. Changing a function on a null set leaves its Lebesgue integral unchanged.
Everywhere: A property that holds at every point in a specified domain. Unlike almost-everywhere truth, it allows no exceptional points.
Measure-zero set: A measurable set whose measure is zero. The exceptional points in an almost-everywhere statement must lie in such a set.
Essential supremum: The least bound that a measurable function satisfies outside a set of measure zero. It ignores exceptional values in the same way almost-everywhere equality does.
Lp space: A space of measurable functions whose absolute values have finite p-th power integral, with functions identified almost everywhere. Its elements are equivalence classes, not individual pointwise function values.
Almost surely: A probability statement that holds with probability one. It is the probability-theoretic name for holding almost everywhere under a probability measure.
Measurable set: A set included in the measurable collection of a measure space, so its measure is defined. Failure sets must be measurable, or otherwise be contained in a measurable null set, for the claim to be well formed.
Equivalence class: A collection of objects identified as equivalent under a specified relation. Function spaces often identify functions that differ only on a null set.

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