Continuous doubling weight vanishing on set of positive measure?

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

A doubling measure on 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 weight is an L1loc function w:RR0 such that wdx is a doubling measure on R.

Question: Is there a continuous doubling 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 Analysis an 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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