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.)

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