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# almost everywhere

Let $(X,\mathfrak{B},\mu)$ be a measure space. A condition holds *almost everywhere* on $X$ if it holds “with probability $1$,” i.e. if it holds everywhere except for a subset of $X$ with measure $0$. For example, let $f$ and $g$ be nonnegative functions on $X$. Suppose we want a sufficient condition on functions $f(x)$ and $g(x)$ such that the relation

$\int_{{X}}fd\mu(x)\leq\int_{{X}}gd\mu(x)$ | (1) |

holds. Certainly $f(x)\leq g(x)$ for all $x\in X$ is a sufficient condition, but in fact it’s enough to have $f(x)\leq g(x)$ almost surely on $X$. In fact, we can loosen the above non-negativity condition to only require that $f$ and $g$ are almost surely nonnegative as well.

If $X=[0,1]$, then $g$ might be less than $f$ on the Cantor set, an uncountable set with measure $0$, and still satisfy the condition. We say that $f\leq g$ almost everywhere (often abbreviated *a.e.*).

Note that this term is the equivalent of the term “almost surely” from probabilistic measure theory.

## Mathematics Subject Classification

60A10*no label found*

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## Corrections

Measure space, classification by Koro ✓

Quotes by bbukh ✓

notation by Mathprof ✘