Corollary of Lebesgue decomposition theorem and counter-example

Question

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 $x$?)