How Does Regularizing Measures Relate to Rudin's Analysis?

[lark]

Reading Rudin's Real and Complex Analysis, a question in the .pdf attached.
(and no pressure about using the built-in Latex, please)
Laura

 

[morphism]

Can you explain how you're "regularizing" $\mu$?

And I suppose a Borel measure on a $\sigma$-compact space like the complex plane has to be regular then?

No. There are examples of Borel measures on $\sigma$-compact (even on compact) spaces which fail to be regular.

 

[lark]

Can you explain how you're "regularizing" $\mu$?

$\mu$ is associated with a bounded linear functional by $\Phi(f) = \int_Xfd\mu$. Then by the Riesz representation theorem, if X is locally compact & Hausdorff, $\Phi$ is associated with a regular measure $\mu^\prime$ by $\Phi(f) = \int_Xfd\mu^\prime$. So $\int_Xfd(\mu-\mu^\prime)=0,$ all $f$ in $C_0(X)$.
So the question is, how much does this say about the measure $\mu-\mu^\prime$? Under what circumstances is it 0, so that all the complex measures on X are regular?

No. There are examples of Borel measures on $\sigma$-compact (even on compact) spaces which fail to be regular.

If X is locally compact and Hausdorff can this still happen? example?

 

[lark]

The argument in Rudin's RACA for why the regular complex measure in the Riesz Representation theorem is unique, does apply just to regular measures. That's because the theorem that $C_c(X)$ (continuous fctns on X with compact support) is dense in $L^p(\mu)$ for $1\le p$< $\infty$ just applies to regular measures.
From a theorem in Rudin's RACA, if $X$ is locally compact and Hausdorff, and every open set is $\sigma$-compact, and $\mu$ is a complex Borel measure on $X$, $|\mu|$ is regular.
So I'm still wondering about regularizing complex measures, as described above, so that you get a new regular complex measure which gives the same integrals on $C_0(X)$. How similar is the new measure? any examples of what happens with this process?
Laura

 

[morphism]

If X is locally compact and Hausdorff can this still happen? example?

The standard example is X=[0,w] where w is the first uncountable ordinal. This is an exercise in Rudin (last one in chapter 2 if you have the first edition).

As for your other question, I don't really know what happens in general. I'll think about it some more and let you know if I come up with anything.

 

[lark]

As for your other question, I don't really know what happens in general. I'll think about it some more and let you know if I come up with anything.

Yeah, I was wondering if regularizing a measure this way is something that's mathematically useful, that you would get a measure that gives the same integrals (a lot of the time at least) but is better behaved. In the usual sensible spaces, all the complex measures are regular anyway, so maybe the measures that aren't regular are mostly weird counterexamples that people don't care about regularizing.
Laura

 

[lark]

see attached .pdf
Laura