Let be a measure space. A condition holds almost everywhere on if it holds “with probability ,” i.e. if it holds everywhere except for a subset of with measure . For example, let and be nonnegative functions on . Suppose we want a sufficient condition on functions and such that the relation
holds. Certainly for all is a sufficient condition, but in fact it’s enough to have almost surely on . In fact, we can loosen the above non-negativity condition to only require that and are almost surely nonnegative as well.
Note that this is the of the “almost surely” from probabilistic measure .
|Date of creation||2013-03-22 12:20:58|
|Last modified on||2013-03-22 12:20:58|
|Last modified by||mathcam (2727)|