2
$\begingroup$

As stated in this question, Lebesgue differentiation theorem holds on locally doubling space? and proved here, http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf the Lebesgue differentiation theorem (LDT) holds in doubling measure spaces. It is also known that every complete doubling metric supports a doubling measure: http://www.ams.org/journals/proc/1998-126-02/S0002-9939-98-04201-4/S0002-9939-98-04201-4.pdf

[From this I conclude that if $\mu$ is a probability measure on a complete doubling metric space $(X,d)$ and $\mu\ll\nu$, where $\nu$ is a doubling measure (whose existence is guaranteed), then $\mu$ is itself a doubling measure and hence satisfies the LDT. EDIT: this is false, ignore.]

Does the LDT hold for probability measures on complete doubling metric spaces? (Can additionally assume $(X,d)$ is compact if needed.)

$\endgroup$
  • 2
    $\begingroup$ I don't think I agree with the second paragraph. Let $X = [0,1]$ with its usual metric, let $\nu$ be Lebesgue measure (which is doubling) and let $\mu$ be the probability measure with cdf $F(x) = e^{1-1/x}$. Then $\mu \ll \nu$ but $\mu$ is not doubling. $\endgroup$ – Nate Eldredge Jul 19 '16 at 21:08
  • $\begingroup$ Ack, I see my mistake. Do you agree that the 2nd paragraph holds if I additionally assume $\nu\ll\mu$? $\endgroup$ – Aryeh Kontorovich Jul 19 '16 at 21:15
  • $\begingroup$ I believe my counterexample still applies. Note that the density $d\mu/d\nu$ is $F'(x)$ which is a.e. strictly positive. $\endgroup$ – Nate Eldredge Jul 19 '16 at 21:16
  • $\begingroup$ Oh I see. I had mistakenly assumed that $F'(x)$ blows up at $0$, but now I see this isn't the case. I'll edit the question to remove that paragraph -- but the main question remains for now. $\endgroup$ – Aryeh Kontorovich Jul 19 '16 at 21:26
  • $\begingroup$ Right. It's true that $d\nu/d\mu = 1/F'$ blows up near 0 but that's not a problem, it's still in $L^1(\mu)$. $\endgroup$ – Nate Eldredge Jul 19 '16 at 21:33
3
$\begingroup$

A version of Lebesgue differentiation for non-doubling measures on doubling metric space appears in section 3 of:

T. Hytonen, A FRAMEWORK FOR NON-HOMOGENEOUS ANALYSIS ON METRIC SPACES, AND THE RBMO SPACE OF TOLSA http://www.raco.cat/index.php/PublicacionsMatematiques/article/download/191394/387513

I leave it to your judgment whether this is of use to you.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.