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

A

doubling measureon R is a (non-trivial, locally finite, Borel) measure μ such that μ(2I)≤Cμ(I) for all intervals I and some fixed constant C.A

doubling weightis an L1loc function w:R→R≥0 such that wdx is a doubling measure on R.

Question:Is there acontinuousdoubling weight on R that vanishes on a set of positive Lebesgue measure?Remarks:

- There are many singular doubling measures on 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 Analysisan example (attributed to Journe) is given of a disjoint partition of R into sets E1 and E2 of positive measure such that the indicator functions χE1 and χE2 are both doubling weights. But this does not answer my question.

**Answer**

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