Re: A measure theory question

In article <j8pej2dlnmg9ucb5kg1v4h7iois6jqi5jc@xxxxxxx>,
David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
On Wed, 18 Oct 2006 13:06:51 -0400, "G. A. Edgar"
<edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <7zgZg.9439$cH6.7192@trnddc07>, TCL <tlim1@xxxxxxxxxxx>
wrote:

I need to see an example of two measures mu, and lambda, such that (i) mu is
not sigma-finite, (ii) lambda is a finite real (or complex) measure, and
(iii) lambda cannot be decomposed into a sum of two measures lambda_1,
lambda_2, with lambda _1 << mu and lambda_2 T mu; i.e. the Lebesgue
decomposition theorem fails for lambda. (In other words, I want to know if
the condition of sigma-finite-ness is necessary in the theorem.)

Thanks.



How about lambda = Lebesgue measure on [0,1] and mu=counting measure.

That, or something along those lines, was my first thought.
But lambda << mu here.

When it comes to non-sigma-finite measures, it is not
sufficient just to have the measure defined on an
arbitrary sigma-field, but to have the property that
if a set has a measurable intersection with each set
of finite measure, it is measurable, and its measure
is the supremum of the measures of sets of finite
measure contained in it. Counting measure on Borel
(or Lebesgue) sets does not satisfy this, and the
necessary extension would be to all sets of reals.
Uncountable discrete measures satisfy the R-N property
with this condition, as do all Borel measures on
locally compact Hausdorff spaces.


(It _is_ a counterexample to the Radon-Nikodym theorem,
which is why it seemed to me at first like an answer
to the question: there is certainly no f such that
d lambda = f d mu. But that's not actually the definition
of "<<" - it's R-N's fault that that's what I think of
when I think of "<<".)


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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