Take the 2-minute tour ×  Let $f \in L^1_{loc}$. We call $x \in
\mathbb R^n$ an Lebesgue point, if    1. $\lim \limits_{R
\rightarrow 0} \frac{1}{m(B_R(x))}\int_{B_R(x)} f \;\;\;\;$
exists     1. $\lim \limits_{R \rightarrow 0} \;\; \sup
\limits_{ B, B' \in \mathcal B_R(x)} \vert
\dfrac{1}{m(B)}\int_{B} f - \dfrac{1}{m(B')}\int_{B'} f \vert =
0$, where $\mathcal B_R(x)$ denotes all balls that contain $x$
with radius at most $R$  Most calculus books state only one of
these definitions, and whereas they state several conclusions or
variations of the theorem, but they usually stick to one of the
above definitions.  However, although the equivalence is
obvious, and indeed trivial from 2. to 1., I do not manage to
prove the implication from 1. to 2. - can someone please explain
me, how this is done?  Edit: I should clarify, that I am aware
both definitions are in general different for, say, a continuous
function with a jump. Nevertheless, for example in Wikipedia I
have encountered both defintions entitled as Lebe