Lebesgue differentiation theorem

In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for Henri Lebesgue.

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  to the Lebesgue integral of ${\displaystyle f\cdot \mathbf {1} _{A}}$, where ${\displaystyle \mathbf {1} _{A}}$ denotes the characteristic function of the set A. It is usually written

with λ the n–dimensional Lebesgue measure.

The derivative of this integral at x is defined to be

where |B| denotes the volume (i.e., the Lebesgue measure) of a ball B  centered at x, and B → x means that the diameter of B  tends to 0.
The Lebesgue differentiation theorem (Lebesgue 1910) 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:

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