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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 ]
We start with a measure space (E, X, μ) where E is a set, X is a σ-algebra of subsets of E, and μ is a (non-negative) measure on E defined on the sets of X. For example, E can be Euclidean n-space R n or some Lebesgue measurable subset of it, X is the σ-algebra of all Lebesgue measurable subsets of E, and μ is the Lebesgue measure.
Suppose a and b are constant, and that f(x) involves a parameter α which is constant in the integration but may vary to form different integrals. Assume that f(x, α) is a continuous function of x and α in the compact set {(x, α) : α 0 ≤ α ≤ α 1 and a ≤ x ≤ b}, and that the partial derivative f α (x, α) exists and is
More generally, the measure μ is assumed to be locally finite (rather than finite) and F(x) is defined as μ((0,x]) for x > 0, 0 for x = 0, and −μ((x,0]) for x < 0. In this case μ is the Lebesgue–Stieltjes measure generated by F. [17] The relation between the two notions of absolute continuity still holds. [18]
Substituting x by ... is a Lebesgue measurable function on ... Radon-Nikodym derivative of the pushforward with respect to Lebesgue measure: ...
An example of this is given [3] by the derivative g of the (differentiable but not absolutely continuous) function f(x) = x 2 ·sin(1/x 2) (the function g is not Lebesgue-integrable around 0).
When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. ... Lebesgue integration, ...
Lebesgue integration; ... the logarithmic derivative of a function f is defined by the formula ... (x) of a real variable x, and takes real, ...