User:Tkojar/Sandbox/Lebesgue Differentiation Theorem

Statement
For a Lebesgue integrable real or complex-valued function f on Rn, the indefinite integral is a set function which maps a measurable set A&thinsp; to the Lebesgue integral of $f \cdot \mathbf{1}_A$, where $\mathbf{1}_{A}$ denotes the characteristic function of the set A. It is usually written


 * $ \displaystyle A \mapsto \int_{A}f\ \mathrm{d}\lambda,$

with λ the n–dimensional Lebesgue measure.

The derivative of this integral at x is defined to be


 * $\displaystyle \lim_{B \rightarrow x} \frac{1}{|B|} \int_{B}f \, \mathrm{d}\lambda,$

where |B| denotes the volume (i.e., the Lebesgue measure) of a ball B&thinsp; centered at x, and B → x means that the diameter of B&thinsp; tends to 0.

The Lebesgue differentiation theorem states that this derivative exists and is equal to f(x) at almost every point x ∈ Rn.

In fact a slightly stronger statement is true. Note that:


 * $\displaystyle \left|\frac{1}{|B|} \int_{B}f(y) \, \mathrm{d}\lambda(y) - f(x)\right| = \left|\frac{1}{|B|} \int_{B}(f(y) - f(x))\, \mathrm{d}\lambda(y)\right| \le \frac{1}{|B|} \int_{B}|f(y) -f(x)|\, \mathrm{d}\lambda(y).$

The stronger assertion is that the right hand side tends to zero for almost every point x. The points x for which this is true are called the Lebesgue points of f.