In mathematics, given a locally Lebesgue integrable function on , a point in the domain of is a Lebesgue point if[1]
Here, is a ball centered at with radius , and is its Lebesgue measure. The Lebesgue points of are thus points where does not oscillate too much, in an average sense.[2]
The Lebesgue differentiation theorem states that, given any , almost every is a Lebesgue point of .[3]
^Bogachev, Vladimir I. (2007), Measure Theory, Volume 1, Springer, p. 351, ISBN 9783540345145.
^Martio, Olli; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard (2008), Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, p. 105, ISBN 9780387855882.
^Giaquinta, Mariano; Modica, Giuseppe (2010), Mathematical Analysis: An Introduction to Functions of Several Variables, Springer, p. 80, ISBN 9780817646127.
mathematics, given a locally Lebesgue integrable function f {\displaystyle f} on R k {\displaystyle \mathbb {R} ^{k}} , a point x {\displaystyle x} in the...
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable...
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension...
graph of that function and the X axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of...
condition than Lebesgue-integrability. The converse does not hold; not all Lebesgue-integrable functions are Riemann integrable. The Lebesgue–Vitali theorem...
Lebesgue space may refer to: Lp space, a special Banach space of functions (or rather, equivalence classes of functions) Standard probability space, a...
mathematics, Lebesgue integrability may refer to: Whether the Lebesgue integral of a function is defined; this is what is most often meant. The Lebesgue integrability...
monotonically grow. It is also called the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase...
equilibrium point Ideal point Inflection point Integral point Isolated point Generic point Heegner point Lattice hole, Lattice pointLebesguepoint Midpoint...
real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability...
Rj,ε)f → 0 at each Lebesguepoint of f; while (Rj − RjTε)f → 0 at each Lebesguepoint of Rjf. So Rj,εf → Rjf on the common Lebesgue points of f and Rjf...
integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated...
In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given...
\|} stands for the Lebesgue measure. This λ {\displaystyle \lambda } is called the intensity of the point process. A stationary point process on R d {\displaystyle...
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be...
early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more general...
definition of the Lebesgue integral for a non-negative function). Remark 4. The proof below does not use any properties of the Lebesgue integral except...
In mathematics, in the area of potential theory, a Lebesgue spine or Lebesgue thorn is a type of set used for discussing solutions to the Dirichlet problem...
If a Poisson point process has a parameter of the form Λ = ν λ {\textstyle \Lambda =\nu \lambda } , where ν {\textstyle \nu } is Lebesgue measure (that...
limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary...
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