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 Almost everywhere - Definition 

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

<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 : RR 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 yf(x,y) is Borel measurable.

In probability theory, the phrases become almost surely, almost certain or almost always, corresponding to a probability of 1.


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