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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 ∈ R n . [ 1 ]
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form () (,), where < (), < and the integrands are functions dependent on , the derivative of this integral is expressible as (() (,)) = (, ()) (, ()) + () (,) where the partial derivative indicates that inside the integral, only the ...
One result on the differentiation of integrals is the Lebesgue differentiation theorem, as proved by Henri Lebesgue in 1910. Consider n-dimensional Lebesgue measure λ n on n-dimensional Euclidean space R n.
More generally, a function is locally (meaning on every bounded interval) absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure. If absolute continuity holds then the Radon–Nikodym derivative of μ is equal almost everywhere to the derivative of F. [16]
The Lebesgue integral is obtained by slicing along the y-axis, using the 1-dimensional Lebesgue measure to measure the "width" of the slices. Folland (1999) summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral of f , one partitions the domain [ a , b ] into subintervals", while in the ...
Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure. Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of R n. If A is a Lebesgue-measurable set with λ(A) = 0 (a null set), then every subset of A is also a null set. A fortiori, every subset of A is measurable.
In other words, if a real function F on [a, b] admits a derivative f(x) at every point x of [a, b] and if this derivative f is Lebesgue integrable on [a, b], then [11] () = (). This result may fail for continuous functions F that admit a derivative f ( x ) at almost every point x , as the example of the Cantor function shows.
If is a Lebesgue measurable function on ), then is Lebesgue ... Radon-Nikodym derivative of the pullback with respect to Lebesgue measure: ...