Re: extension of measure

On Aug 3, 8:17 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Sun, 2 Aug 2009 14:59:24 -0700 (PDT), Hannah

<sempre.do.meu.cora...@xxxxxxxxxxxxxx> wrote:
Could you please give me an example of a countably additive measure on
an algebra F

"countably additive measure on an algebra F" is not very clear.
Presumably you mean that mu(union A_j) = sum mu(A_j)
whenever the A_j are in F and it also happens that the union
of the A_j is in F? Such a thing is often called a "premeasure".

that has more than one extension that is a measure?

Say F_1 is a sigma-algebra, mu is a measure on F_1,
F_2 is a sub-sigma-algebra of F_2 and F_3 is a
sub-sigma-algebra of F_1. Then the restriction of
mu to F_3 has more than one proper extension to a
measure, namely the restriction to F_1 and the
restriction to F_2.

That's probably not an example of what you really
want, but it's an example of what you asked for...
(Can you state what you really want more precisely?)

(one should come from the outer measure i think, i still can't think
of an example...)

Thanks

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

Thanks for your answer. What I wanted is a concrete example of an
algebra F on a set X and a countably additive measure mu on F (which
means what you assumed meant was), such that there is more than one
extension of mu to sigma(F) -so this means that mu is not sigma-
finite. One extension should come from the outer measure (analogously
to the extension theorem of caratheodory).
Could you help me construct and understand such an example?
.

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