# Properties of Lebesgue points of a function

Let $$F:\mathbb{R}^2 \rightarrow \mathbb{R}$$ be such that for every $$a,b \in \mathbb{R}$$ the fuctions $$F(a,\cdot), F(\cdot,b) \in L^1(\mathbb{R})$$.

Consider the set $$A:= \left\{a:\lim\limits_{r \rightarrow 0}\frac{1}{\mu{(B(a,r))}}\int_{B(a,r)}F(a,y) d y = F(a,a) \right\}$$, where $$\mu$$ is the Legesgue measure on $$\mathbb{R}$$.

Is $$\mu(\mathbb{R}/A)=0$$?

P.S. By Lebesgue differentiation theorem for every $$a\in \mathbb{R}$$ the set of all Legesgue points of the function $$F(a,\cdot)$$ denoted by $$A_a$$ is measurable and satisfies $$\mu (\mathbb{R} / A_a)=0$$ .

The set $$A$$ defined above is also given by $$A= \cup_{a\in \mathbb{R}} (A_a \cap \{a \})$$. But I am unable to conclude the value of $$\mu(\mathbb{R}/A)$$

Any help would be greatly appreciated. Thanks in advance.

• How is your $A_a$ defined? Nov 20 at 19:20
• Thanks. Updated the question Nov 21 at 12:34

Try $$F(x,y) = 0$$ for $$x\ne y$$, $$1$$ for $$x=y$$.
• @Veronica I think you get that $A=\emptyset$ in this example, which is why its a counter example to your statement. Nov 21 at 13:26