The statement of Lebesgue differentiation theorem is:
For any locally integrable function $f$ on $\mathbb{R}^n$ we have $$ f(x)=\lim_{r\to 0^+}\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)dy.\tag{*} $$ for almost all $x$ in $\mathbb{R}^n$.
In the proof of Lebesgue differentiation theorem most of the books, for example "Stein, Elias M.; Shakarchi, Rami (2005). Real analysis.", says that:
It suffices to show that for each $\lambda>0$ the set
$$ \left\{x:\limsup_{r\to 0}\left|\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)dy-f(x)\right|>\lambda\right\} $$ has measure zero. My questions are:
1) Why proofs don't show the existence of the limit in (*)?
2) According to the proofs something is proven more general by using $\limsup$ instead of $\lim$. Why the statement of the theorem never given by using $\limsup$?
Let $$ \lim_{r\to 0^+}\frac{1}{|B(x,r)|}\int_{B(x,r)}|f(y)-f(x)|dy=0.\tag{**} $$
I read somewhere that $*\Leftrightarrow **$. The implication $**\Rightarrow *$ is trivial but how can we prove that $*\Rightarrow **$.
My last question is how can we generalize (**) to the case $$ \lim_{r\to 0^+}\frac{1}{|B(x,r)|}\int_{B(x,r)}|f(y)-f(x)|^p dy=0, \qquad p>1 $$ for almost all $x$ in $\mathbb{R}^n$.