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Almost everywhere

In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i.e. is a set with measure zero. If used for properties of the real numbers, the Lebesgue measure is assumed unless otherwise stated.

Occasionaly, instead of saying that a property holds almost everywhere, one also says that the property holds for almost all elements. The term almost all in addition has several other meanings however.

Here is a list of theorems that involve the term "almost everywhere":

• If f : R -> R is a Lebesgue integrable function and f(x) ≥ 0 almost everywhere, then ∫ f(x) dx ≥ 0.
• If f : [a, b] -> R is a monotonic function, then f is differentiable almost everywhere.
• If f : R → R is Lebesgue measurable and ∫_a^b|f(x)|dx<∞ for every real numbers a<b then there exists a null set E (depending on f) such that, if x is not in E, the Lebesgue mean 1/(2e)∫_x-e^x+ef(t)dt converges to f(x) as e decreases to zero. In other words, the Lebesgue mean of f converges to f almost everywhere. The set E is called the Lebesgue set of f.
• If f(x,y) is Borel measurable on R² then for almost every x, the function y→f(x,y) is Borel measurable.