Almost everywhere
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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.
Occasionally, instead of saying that a property holds almost everywhere, one also says that the property holds for almost all elements, though the term almost all also has other meanings.
Here are some theorems that involve the term "almost everywhere":
- If f : R → R is a Lebesgue integrable function and f(x) ≥ 0 almost everywhere, then
- <math>\int f(x) \, dx \geq 0.<math>
- If f : [a, b] → R is a monotonic function, then f is differentiable almost everywhere.
- If f : R → R is Lebesgue measurable and
- <math>\int_a^b |f(x)| \, dx < \infty<math>
- 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
- <math>\frac{1}{2e} \int_{x-e}^{x+e} f(t)\,dt<math>
- 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.
In probability theory, the phrases become almost surely, almost certain or almost always, corresponding to a probability of 1.