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#measure-theory

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n -dimensional Euclidean space Rn . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1.

Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ -algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.[1] Indeed, their existence is a non-trivial consequence of the axiom of choice

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Measure (mathematics) - Wikipedia
[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma




#measure-theory
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
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In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns th

Original toplevel document

Measure (mathematics) - Wikipedia
[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma




Flashcard 1732833053964

Tags
#measure-theory
Question
a measure on a set is a systematic way to assign a number to [...], intuitively interpreted as its size.
Answer
each suitable subset of that set

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In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

Original toplevel document

Measure (mathematics) - Wikipedia
[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma







Qudama b. Jacfar (d. after 320/932), in his Naqd al-shier, says explicitly that a poet should not be blamed for contra- dicting himself in his verse, and that the quality of his poetry is not affected by obscene or 'scandalous' themes (al-macna 1-fabish, fabashat al-macna). 38 c Ali b. c Abd ai-c Aziz al-Jurjani (d. 392/1001), although a qar;IJ, wrote in his book on the poetry of al-Mutanabbi that 'religion is detached from poetry' (al-din bi-maczil can al-shicr). 39 The Qur 0 an says that poets 'say things they do not do' (Q 26:226), which has been exploited by poets (including al-Farazdaq) 40 to exculpate themselves on occasion.
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Flashcard 1733044604172

Tags
#redes #tcp-ip
Question
TCP/IP. Características Básicas
Answer
Los servicios que TCP proporciona para la comunicación entre aplicaciones, pueden caracterizarse en los cuatro puntos siguientes:
- Cuando dos aplicaciones transfieren grandes cantidades de datos, podemos pensar en esos datos como secuencias de octetos. El módulo TCP del nodo destino entrega las secuencias de octetos que recibe a la capa superior, tal y como la capa superior del nodo origen se la entregó a TCP.
- Circuito virtual: TCP proporciona un servicio de comunicación orientado a la conexión. De esta forma TCP suministra un servicio fiable de transporte de datos de extremo a extremo. Entre otros servicios, TCP proporciona el establecimiento de la conexión, la transferencia de información y la desconexión.
- Buffers de datos. Las aplicaciones transfieren sus datos en forma de cadenas de octetos al módulo TCP. Este agrupa o divide las cadenas de octetos recibidas en función del tamaño del paquete que puede transmitir. Para hacer más eficiente la transmisión y reducir el tráfico en la red, es necesario que el módulo TCP disponga de buffers donde acumular la información. Obviamente estos buffers pueden retrasar la transmisión de determinada información. Cuando es necesario realizar la transmisión inmediata de algunos datos, puede solicitarse a TCP que transmita dicha información (push), aún cuando su tamaño sea menor que el que proporciona mayor rendimiento.
- Conexión full-duplex: |TCP/IP permite que las conexiones establecidas entre dos nodos sean concurrentes o simultáneas en ambos sentidos. Las aplicaciones ven este servicio como dos canales de transmisión (uno de entrada y otro de salida), sin interacción aparente.

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Características Básicas. Los servicios que TCP proporciona para la comunicación entre aplicaciones, pueden caracterizarse en los cuatro puntos siguientes: - Cuando dos aplicaciones transfieren grandes cantidades de datos, podemos pensar en esos datos como secuencias de octetos. El módulo TCP del nodo destino entrega las secuencias de octetos que recibe a la capa superior, tal y como la capa superior del nodo origen se la entregó a TCP. - Circuito virtual: TCP proporciona un servicio de comunicación orientado a la conexión. De esta forma TCP suministra un servicio fiable de transporte de datos de extremo a extremo. Entre otros servicios, TCP proporciona el establecimiento de la conexión, la transferencia de información y la desconexión. - Buffers de datos. Las aplicaciones transfieren sus datos en forma de cadenas de octetos al módulo TCP. Este agrupa o divide las cadenas de octetos recibidas en función del tamaño del paquete que puede transmitir. Para hacer más eficiente la transmisión y reducir el tráfico en la red, es necesario que el módulo TCP disponga de buffers donde acumular la información. Obviamente estos buffers pueden retrasar la transmisión de determinada información. Cuando es necesario realizar la transmisión inmediata de algunos datos, puede solicitarse a TCP que transmita dicha información (push), aún cuando su tamaño sea menor que el que proporciona mayor rendimiento. - Conexión full-duplex: |TCP/IP permite que las conexiones establecidas entre dos nodos sean concurrentes o simultáneas en ambos sentidos. Las aplicaciones ven este servicio como dos canales de transmisión (uno de entrada y otro de salida), sin interacción aparente.

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