Search results
Results From The WOW.Com Content Network
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 ]
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. In the mathematical theory of probability, we confine our study to a probability measure μ , which satisfies μ ( E ) = 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 ...
The absolute value function : +, = | |, which is not differentiable at = has a weak derivative : known as the sign function, and given by () = {>; =; < This is not the only weak derivative for u: any w that is equal to v almost everywhere is also a weak derivative for u. For example, the definition of v(0) above could be replaced with any ...
Legendre functions are solutions of Legendre's differential equation (generalized or not) with non-integer parameters. In physical settings, Legendre's differential equation arises naturally whenever one solves Laplace's equation (and related partial differential equations) by separation of variables in spherical coordinates.
The result for Lebesgue measure turns out to be a special case of the following result, which is based on the Besicovitch covering theorem: if μ is any locally finite Borel measure on R n and f : R n → R is locally integrable with respect to μ, then (()) () = for μ-almost all points x ∈ R n.
Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functions, whose set of non-differentiability points must be a Lebesgue null set (Rademacher's theorem). When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz or otherwise well-behaved.
The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of ...