The notion of a point of vanishing mean oscillation is more general than that of Lebesgue points. How can I demonstrate that the notion of a vanishing mean oscillation point (VMO) is more general than a Lebesgue point?
The point $y\in\mathbb{R}^n$ is a Lebesgue point if:
$$ \lim_{r\to 0} \oint_{B_r(y)} |f(x)-f(y)|\;  dx = 0 $$
The point $y\in\mathbb{R}^n$ is a VMO point (vanishing mean oscillation) if:
$$ \lim_{r\to 0} \oint_{B_r(y)} |f(x)-(f)_{y,r}|\;  dx  = 0 $$
Here $\oint_U f=\frac{1}{|V|}\int_V f$ is the average integral of $f$ (I don't know the command in LateX to write an integral with a dash in the middle).
And $(f)_{y,r} = \oint_{B_r(y)}$.

I tried to put the limit inside the integral. But I can't see that it generalizes.
Some references
VMO1: p. 2 of https://www.ams.org/journals/tran/1975-207-00/S0002-9947-1975-0377518-3/S0002-9947-1975-0377518-3.pdf
VMO 2: http://en.wikipedia.org/wiki/Bounded_mean_oscillation
Lebesgue point: https://en.wikipedia.org/wiki/Lebesgue_point
 A: $\def\avint{\mathop{\rlap{\raise.15em{\scriptstyle --}}\kern-.13em\int}\nolimits}$
If $y$ is a Lebesgue point, it is easy to see that $f_{y,r} \to f(y)$ when $r \to 0$:
\begin{align}
|f_{y,r} - f(y)| 
&= \left| \avint_{B_r(y)} f(x) dx - f(y) \right| \\
&= \left| \avint_{B_r(y)} \left( f(x) - f(y) \right) dx \right| \\
& \leqslant \avint_{B_r(y)} |f(x) - f(y)| dx \\
&\to 0.
\end{align}
By triangle inequality, $y$ also satisfies the vanishing mean oscillation condition:
\begin{align}
\avint_{B_r(y)} |f(x) - f_{y,r}| dx 
&\leqslant \avint_{B_r(y)} |f(x)-f(y)| dx + \avint_{B_r(y)} |f_{y,r}-f(y)| dx \\
&\leqslant \avint_{B_r(y)} |f(x)-f(y)| dx + |f_{y,r}-f(y)| \\
&\to 0.
\end{align}
A: Using the  Lebesgue differentiation Theorem we know that:
$$ \lim_{r\to 0+}\left(\frac{1}{|B_r(y)|}\int_{B_r(y)} f(x)\, dx \right) = f(y)\; a.e \; y\in \mathbb{R}^n $$
Then
$$ \oint_{B_r(y)} |f(x)-(f)_{r,y} |\; dx \leq \oint_{B_r(y)} |f(x)-f(y)|\, dx + \oint_{B_r(y)} |f(y)-(f)_{r,y} | \to 0$$
Where the first installment of the sum goes to 0 if $y$ is a Lebesgue point and the second, is the Lebesgue Differentiation Theorem above.
