| In a measure space a property is said to hold "almost
everywhere" if the set of exceptions has measure zero. For
sets like R (the real numbers) or C (the complex numbers) or
R^n or C^n, the measure is usually taken to be the Lebesgue
measure, although the context may indicate otherwise.
For cardinals, especially regular cardinals, one commonly
defines a property to hold "almost everywhere" if the set of
exceptions has smaller cardinality than the cardinal itself.
So a property of integers holds "almost everywhere" if there
are at most finitely many exceptions, or a property of the
elements of Omega-1 holds "almost everywhere" if there are at
most countably many exceptions.
In the most general case one has a set X and an ideal I of
subsets of X considered "small", and a property of elements of
X holds "almost everywhere" if the set of exceptions is an
element of the ideal of "small" subsets of X. (In many cases
the ideal I will not be explicitly mentioned, e.g., when X is
the reals it is commonly assumed that I is the ideal of
subsets of the reals of Lebesgue measure zero.)
Dan
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| Sometimes the French "p.p." (presque partout) is used instead of a.e.
The point is that if something holds a.e. then you can sort of ignore
those cases where it doesn't hold (lots of hand waving here) because
sets of zero measure don't affect your integration results, for example.
Kind of like saying "The President's new tax plan is fair and equitable
almost everywhere", meaning the rich who will get royally screwed don't
have enough votes to hurt anyone screwing them.
John
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