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...

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

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

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...

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...

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

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...

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

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...

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

the space X {\displaystyle X} the measure μ {\displaystyle \mu } is non-zero. Consider two examples: Lebesgue measure λ {\displaystyle \lambda } on the...

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)...

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...

interval [ 0 , 1 ] {\displaystyle [0,1]} is equal to the restriction of Lebesgue measure to the Borel subsets of R {\displaystyle \mathbb {R} } . This can be...

single point. More formally, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if its support is at most a countable...

distance in terms of elementary probability theory, we take λ to be the Lebesgue measure, so that dP / dλ and dQ / dλ are simply probability density functions...

two Lebesgue measures to give the Lebesgue measure on the product space. The opposite construction to the formation of the product of two measures is disintegration...

regular. The Lebesgue measure on the real line is a regular measure: see the regularity theorem for Lebesgue measure. Any Baire probability measure on any locally...

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...

Lebesgue measure Lebesgue integration Lebesgue's density theorem Counting measure Complete measure Haar measure Outer measure Borel regular measure Radon...

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 ) = |...

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

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...

complete measure spaces is not usually complete. For example, the product of the Lebesgue measure on the unit interval I with itself is not the Lebesgue measure...

cat map, which is ergodic for the Lebesgue measure on the torus. If μ {\displaystyle \mu } is a probability measure on a space X {\displaystyle X} which...

called a σ {\displaystyle \sigma } -finite measure space. For example, Lebesgue measure on the real numbers is not finite, but it is σ-finite. Indeed, consider...

The integral is with respect to an existing measure μ, which may often be the canonical Lebesgue measure on the real line R or the n-dimensional Euclidean...

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