Differentiation of integrals

In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks for what functions f : X → R does lim r → 0 1 μ ( B r ( x ) ) ∫ B r ( x ) f ( y ) d μ ( y ) = f ( x ) {\displaystyle \lim _{r\to 0}{\frac {1}{\mu {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \mu (y)=f(x)} for all (or at least μ-almost all) x ∈ X?

Source: Wikipedia — Differentiation of integrals (CC BY-SA 4.0)

Differentiation of integrals

In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks for what functions f : X → R does lim r → 0 1 μ ( B r ( x ) ) ∫ B r ( x ) f ( y ) d μ ( y ) = f ( x ) {\displaystyle \lim _{r\to 0}{\frac {1}{\mu {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \mu (y)=f(x)} for all (or at least μ-almost all) x ∈ X?

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Source: Wikipedia "Differentiation of integrals" · CC BY-SA 4.0

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