In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe") on a set X...

. Henri Lebesgue, for a non-technical description of Lebesgue integration Null set Integration Measure Sigma-algebra Lebesgue space Lebesgue–Stieltjes...

theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability...

{\displaystyle \Sigma } a σ {\displaystyle \sigma } -algebra over X . {\displaystyle X.} A set function μ {\displaystyle \mu } from Σ {\displaystyle \Sigma } to...

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,...

the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by ( R , B , λ ) ....

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 space Rn (corresponding...

known as the shadow price in economics the Lebesgue measure denotes the volume or measure of a Lebesgue measurable set longitude in geodesy linear density...

transform is both unitary on L2 and an algebra homomorphism from L1 to L∞, without renormalizing the Lebesgue measure. When the independent variable (...

-i\sigma _{1}=-\sigma _{2}\sigma _{3},\quad \mathbf {j} \mapsto -i\sigma _{2}=-\sigma _{3}\sigma _{1},\quad \mathbf {k} \mapsto -i\sigma _{3}=-\sigma _{1}\sigma...

In abstract algebra, the trigintaduonions, also known as the 32-ions, 32-nions, 25-nions, or sometimes pathions ( P {\displaystyle \mathbb {P} } ), form...

was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory. Lebesgue spaces are function spaces...

often appear when working with product (Cartesian) spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection...

is a sub-σ-algebra of F {\displaystyle {\mathcal {F}}} . Since H {\displaystyle {\mathcal {H}}} is a sub σ {\displaystyle \sigma } -algebra of F {\displaystyle...

of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel;...

is not Lebesgue measurable. There exist Lebesgue-measurable subsets of the real numbers that are not Borel sets. That is, the Borel σ-algebra on the real...

Wydawnictwo SIGMA-NOT: 38–41. doi:10.15199/48.2015.02.09. ISSN 0033-2097. Alfsmann, Daniel (2006), "On families of 2^N dimensional hypercomplex algebras suitable...

of bounded variation the corresponding Lebesgue–Stieltjes measure, and the integral with respect to the Lebesgue–Stieltjes measure agrees with the Riemann–Stieltjes...

Elektrotechniczny. 1 (2). Wydawnictwo SIGMA-NOT: 38–41. doi:10.15199/48.2015.02.09. ISSN 0033-2097. Richard D. Schafer (1954) "On the algebras formed by the Cayley–Dickson...

that year and by Hermann Weyl in 1928: σ x σ p ≥ ℏ 2 {\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}} where ℏ = h 2 π {\displaystyle \hbar...

\quad \Sigma =\Sigma _{1}\otimes \Sigma _{2},\quad \mu =\mu _{1}\otimes \mu _{2},} where S is the Cartesian product of S1 and S2, the σ-algebra Σ arises...

k} have algebraic closure K . {\displaystyle K.} If the distinct embeddings of E {\displaystyle E} are { σ j } j , {\displaystyle \left\{\sigma _{j}\right\}_{j}...

{\displaystyle (\sigma _{1},j,k),(i,\sigma _{2},k),(i,j,\sigma _{3}),(\sigma _{1},\sigma _{2},\sigma _{3}),} (★ σ 1 , i , l ) , ( {\displaystyle \sigma _{1},i,l)...

function f : Rn → R. Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of ω to be the integral of...

space (X, Σ) equipped with a sigma-finite measure μ and a transformation τ. Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X, Σ)...

axiomatic set theory. Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration, which proved...

is absolutely continuous with respect to the Lebesgue measure, μsc is singular with respect to the Lebesgue measure and atomless, and μpp is a pure point...

{\displaystyle \sigma (T)=\sigma _{p}(T)\cup \sigma _{c}(T)\cup \sigma _{r}(T).} The complement of the spectrum σ ( T ) {\displaystyle \sigma (T)} is known...

d x {\displaystyle dx} and d F ( x ) {\displaystyle dF(x)} are Lebesgue and Lebesgue–Stieltjes integrals, respectively. If the function x 2 f ( x ) {\displaystyle...

topology Polish Space Cantor space Inductive dimension Lebesgue covering dimension Lebesgue's number lemma Polytope Simplex Simplicial complex CW complex...