In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure...

straightforward way to more general spaces, measure spaces, such as those that arise in probability theory. The term Lebesgue integration can mean either the general...

Lebesgue's density theorem Lebesgue differentiation theorem Lebesgue integration Lebesgue's lemma Lebesgue measure Lebesgue's number lemma Lebesgue point...

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...

In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes both Riemann–Stieltjes and Lebesgue integration...

infinite-dimensional Lebesgue measure is a measure defined on infinite-dimensional normed vector spaces, such as Banach spaces, which resembles the Lebesgue measure used...

the fundamental theorem of Lebesgue integral calculus, due to Lebesgue. For an equivalent definition in terms of measures see the section Relation between...

set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). This is the Lebesgue-Vitali theorem (of characterization of the...

that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai...

Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which...

Lebesgue differentiation theorem Lebesgue integration Lebesgue measure Infinite-dimensional Lebesgue measure Lebesgue point Lebesgue space Lebesgue–Rokhlin...

products. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may...

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)...

variables formula for Lebesgue measure, we have that Radon-Nikodym derivative of the pullback with respect to Lebesgue measure: d T ∗ m d m ( x ) = |...

partially outside of A, is negligible. Let μ be the Lebesgue measure on the Euclidean space Rn and A be a Lebesgue measurable subset of Rn. Define the approximate...

Rokhlin in 1940. Rokhlin showed that the unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, yet can be...

set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance using Carathéodory's...

positive Lebesgue measure. A meagre set in R {\displaystyle \mathbb {R} } need not have Lebesgue measure zero, and can even have full measure. For example...

of integrals is the Lebesgue differentiation theorem, as proved by Henri Lebesgue in 1910. Consider n-dimensional Lebesgue measure λn on n-dimensional...

spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by ( R , B , λ ) . {\displaystyle (\mathbb...

natural "Lebesgue measure" on S1 is then the push-forward measure f∗(λ). The measure f∗(λ) might also be called "arc length measure" or "angle measure", since...

measures: Lebesgue measure on Euclidean space (restricted to the Borel subsets); Haar measure on any locally compact topological group; Dirac measure...

{\displaystyle \mathbb {R} } the Lebesgue σ-algebra of R {\displaystyle \mathbb {R} } , i.e., the set of all Lebesgue measurable sets in R {\displaystyle...

measure μ {\displaystyle \mu } defined on X {\displaystyle X} is an SRB measure if there exist U ⊂ X {\displaystyle U\subset X} of positive Lebesgue measure...

result of Lebesgue given above. Indeed, when s = d, the Hausdorff measure Hs on Rd coincides with a multiple of the d-dimensional Lebesgue measure. If a disjoint...

real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability...

the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also...

appear. For example, Lebesgue measure on the reals is not a probability measure, yet every universally measurable set is Lebesgue measurable. To see this...

measurable subset of R n {\displaystyle \mathbb {R} ^{n}} with the Lebesgue measure, and f {\displaystyle f} and g {\displaystyle g} are measurable real-...

mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated...