If $I$ is a bounded interval in $\mathbb{R}$, let $2I$ denote an interval with the same center point but double the length.

A *doubling measure* on $\mathbb{R}$ is a (non-trivial, locally finite, Borel) measure $\mu$ such that $\mu(2I) \leq C\mu(I)$ for all intervals $I$ and some fixed constant $C$.

A *doubling weight* is an $L^1_{\text{loc}}$ function $w:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ such that $w dx$ is a doubling measure on $\mathbb{R}$.

**Question:** Is there a *continuous* doubling weight on $\mathbb{R}$ that vanishes on a set of positive Lebesgue measure?

Remarks:

- There are many singular doubling measures on $\mathbb{R}$, not arising from doubling weights.
- Without the assumption of continuity, there are doubling weights that vanish on sets of positive Lebesgue measure. In Section 1.8.8 of Stein,
*Harmonic Analysis*an example (attributed to Journe) is given of a disjoint partition of $\mathbb{R}$ into sets $E_1$ and $E_2$ of positive measure such that the indicator functions $\chi_{E_1}$ and $\chi_{E_2}$ are both doubling weights. But this does not answer my question.

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