Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Refferring to the Lebesgue decomposition theorem in Lebesgue decomposition theorem and fundamental theorem of calculus there is a corollary when the measure is the Lebesgue measure that states: if $f\in L^p_{loc}(\mathbb{R}^n)$ then $$\lim \frac{1}{|B|}\int_{B}|f-f(x)|^p=0$$ for a.e. $x$, where the limit is taken on balls B each containing $x$ with diameter tending to zero (x is not necessarily the center of the balls). Is there a counter-example of this fact if we choose a general Radon measure (I know taking the limit on balls centered in x holds the same result, but could this not happen if the balls are not centered in $x$?)

share|cite|improve this question
Looking at the proof I guess maybe is the measure is not doubling this couldn't work, but I don't even know real examples od non-doubling measures.. – balestrav Apr 20 '12 at 13:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.