User:Addemf/sandbox/Length-measure and Differentiation

First Lebesgue Differentiation Theorem
We return to that problem which sent us along this sequence of thoughts, which is the proof that if f is integrable then


 * $$ \lim_{t\to 0^+}\frac{1}{2t}\int_{(x-t,x+t)}|f-f(x)|=0$$

almost everywhere on $$\Bbb R$$.

To begin the proof we let $$A\subseteq \Bbb R$$ be the set of points at which this equality fails. Therefore we intend to show that its measure is zero.

Let $$\delta\in\Bbb R^+$$ and we will attempt to approximate A in such a way that we are able to show $$\lambda(A)$$ is small.

We have shown already that, for each $$k\in\Bbb Z^+$$ there is a continuous function $$\int|f-h_k| < \delta/2^k$$.