Definition of countable base for a measure
sto
I think is not quite correct:
Let (X,S,m) be a sigma-finite measure space. If S contains a countable
class of sets A be having the property that, for any e>0 and E in S,
there exists an E0 in A such that m(E\E0 \/ E0\E) < e, then m is said to
have a *countable base*.
It seems to me that the correct definition is actually "for any e>0 and
E in S having m(E) < oo". Is that not the case?
David C. Ullrich
Definitions are arbitrary - they can't be "correct" or "incorrect".
Even when a term has a standard definition, there's nothing
wrong with a non-standard use of the term, _if_ the
non-standard definition is given explicitly.
The more important question is whether the way the author
of whatever you're reading uses the term in a way that's
consistent with the definition he gives. (Hmm, for example
Lebesgue measure on R has a countable base in the one
sense but not in the other...)
Having said that definitions are arbitrary, I'll also say this:
If I saw this terminology wth no defintion given, my
guess at the meaning would allow m(E) = infinity, but
would also allow E0 to be an element of the sigma-algebra
generated by A.