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Explore the people, places, events, achievements, struggles and stories that shaped our journey.
Movements, leaders, victories and the continuing fight for equality.
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Meet the people whose lives, choices and achievements shaped the journey.
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In August 1908, a white mob attacked Springfield, Illinois’s Black community, destroying homes and businesses and lynching two Black men. National outrage over the violence helped spur the movement that created the NAACP the following year.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e.,
where the functions each have a countable range and for which the pre-image is measurable for each element x. The concept is named after Salomon Bochner.
Bochner-measurable functions are sometimes called strongly measurable, -measurable or just measurable (or uniformly measurable in case that the Banach space is the space of continuous linear operators between Banach spaces).
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.
Function f is almost surely separably valued (or essentially separably valued) if there exists a subset N ⊆ X with μ(N) = 0 such that f(X \ N) ⊆ B is separable.
A function f : X → B defined on a measure space (X, Σ, μ) and taking values in a Banach space B is (strongly) measurable (with respect to Σ and the Borel algebra on B) if and only if it is both weakly measurable and almost surely separably valued.
In the case that B is separable, since any subset of a separable Banach space is itself separable, one can take N above to be empty, and it follows that the notions of weak and strong measurability agree when B is separable.
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Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e., f ( t ) = lim n → ∞ f n ( t ) for almost every t , {\displaystyle f(t)=\lim _{n\rightarrow \infty }f_{n}(t){\text{ for almost every }}t,\,} where the functions f n {\displaystyle f_{n}} each have a countable range and for which the pre-image f n − 1 ( { x } ) {\displaystyle f_{n}^{-1}(\{x\})} is measurable for each element x. The concept is named after Salomon Bochner. Bochner-measurable functions are sometimes called strongly measurable, μ {\displaystyle \mu } -measurable or just measurable (or uniformly measurable in case that the Banach space is the space of continuous linear operators between Banach spaces).
In mathematics, Bochner spaces are a generalization of the concept of L p {\displaystyle L^{p}} spaces to functions whose values lie in a Banach space which is not necessarily the space R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } of real or complex numbers. The space L p ( X ) {\displaystyle L^{p}(X)} consists of (equivalence classes of) all Bochner measurable functions f {\displaystyle f} with values in the Banach space X {\displaystyle X} whose norm ‖ f ‖ X {\displaystyle \|f\|_{X}} lies in the standard L p {\displaystyle L^{p}} space. Thus, if X {\displaystyle X} is the set of complex numbers, it is the standard Lebesgue L p {\displaystyle L^{p}} space. Almost all standard results on L p {\displaystyle L^{p}} spaces do hold on Bochner spaces too; in particular, the Bochner spaces L p ( X ) {\displaystyle L^{p}(X)} are Banach spaces for 1 ≤ p ≤ ∞ . {\displaystyle 1\leq p\leq \infty .} Bochner spaces are named for the mathematician Salomon Bochner.
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.
In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →The Greenwood District.
