# Lebesgue Points Question

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• Mar 10th 2014, 03:53 AM
Johnyboy
Lebesgue Points Question
Let $\displaystyle \Omega \subset \mathbb{R}^{n}$ be a bounded open set and take $\displaystyle p' = \frac{p}{(p-1)}$. Let $\displaystyle a: \Omega \times \mathbb{R} \times \mathbb{R}$ be a Caratheodory function satisfying $\displaystyle |a(x,s,\xi)| \leq k(x) + \beta(|s|^{p-1} + |\xi|^{p-1})$ for almost every $\displaystyle x \in \Omega$, for every $\displaystyle (s, \xi) \in \mathbb{R} \times \mathbb{R}^{n}$ and for some $\displaystyle k \in L^{p^{'}}(\Omega), \beta \geq 0, p > 1$.

If we further assume that $\displaystyle x_{0}$ is a Lebesgue point of $\displaystyle a$ and let $\displaystyle s \in \mathbb{R}$, $\displaystyle \xi, \eta \in \mathbb{R}^{n}$ ($\displaystyle \xi \neq \eta$) be such that $\displaystyle \langle a(x_{0},s,\xi)-a(x_{0},s,\eta); \xi-\eta \rangle < 0$. Then how does it follow that since $\displaystyle x_{0}$ is a Lebesgue point of $\displaystyle a$ gives us

$\displaystyle \lim\limits_{R \rightarrow 0}\{\frac{1}{\text{meas}H_{R}(x_{0})}}\int_{H_{R}( x_{0})}\langle a(x,s,\xi)-a(x,s,\eta); \xi-\eta \rangle dx \}< 0$, where $\displaystyle H_{R}(x_{0})$ is Hypercube of $\displaystyle \mathbb{R}^{n}$?

Let me know if any further info is required. Thanks.