measure

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measure

from "Roses of the South," a waltz by Johann Strauss the Younger

1. Dimensions, quantity, or capacity as ascertained by comparison with a standard.
2. A reference standard or sample used for the quantitative comparison of properties: The standard kilogram is maintained as a measure of mass.
3. A unit specified by a scale, such as an inch, or by variable conditions, such as a day's march.
4. A system of measurement, such as the metric system.
5. A device used for measuring.
6. The act of measuring.
7. An evaluation or a basis of comparison: "the final measure of the worth of a society" (Joseph Wood Krutch). See synonyms at standard.
8. Extent or degree: The problem was in large measure caused by his carelessness.
9. A definite quantity that has been measured out: a measure of wine.
10. A fitting amount: a measure of recognition.
11. A limited amount or degree: a measure of good-will.
12. Limit; bounds: generosity knowing no measure.
13. Appropriate restraint; moderation: "The union of . . . fervor with measure, passion with correctness, this surely is the ideal" (William James).
14. An action taken as a means to an end; an expedient. Often used in the plural: desperate measures.
15. A legislative bill or enactment.
16. Poetic meter.
17. Music. The metric unit between two bars on the staff; a bar.

1. To ascertain the dimensions, quantity, or capacity of: measured the height of the ceiling.
2. To mark, lay out, or establish dimensions for by measuring: measure off an area.
3. To estimate by evaluation or comparison: "I gave them an account . . . of the situation as far as I could measure it" (Winston S. Churchill).
4. To bring into comparison: She measured her power with that of a dangerous adversary.
5. 1. To mark off or apportion, usually with reference to a given unit of measurement: measure out a pint of milk.
   2. To allot or distribute as if by measuring; mete: The revolutionary tribunal measured out harsh justice.
6. To serve as a measure of: The inch measures length.
7. To consider or choose with care; weigh: He measures his words with caution.
8. Archaic. To travel over: "We must measure twenty miles today" (Shakespeare).

1. To have a measurement of: The room measures 12 by 20 feet.
2. To take a measurement.
3. To allow of measurement: White sugar measures more easily than brown.

measure up

1. To be the equal of something; have similar quality.
2. To have the necessary qualifications: a candidate who just didn't measure up.

beyond measure

1. In excess.
2. Without limit.

1. In addition to the required amount.

1. To a degree: The new law was in a measure harmful.

[Middle English, from Old French mesure, from Latin mēnsūra, from mēnsus, past participle of mētīrī, to measure.]

A reference sample used in comparing lengths, areas, volumes, masses, and the like. The measures employed in scientific work are based on the international units of length, mass, and time—the meter, the kilogram, and the second—but decimal multiples and submultiples are commonly employed. Prior to the development of the international metric system, many special-purpose systems of measures had evolved and many still survive, especially in the United Kingdom and the United States. See also Metric system; Physical measurement; Units of measurement; Weight.

Width of a line of type defining the number of characters that may be set in the space available; usually expressed in picas (sixths of an inch). A typical book has 40 lines per page, with 50 to 75 characters per line.

noun

1. The amount of space occupied by something: dimension, extent, magnitude, proportion (often used in plural), size. See big/small/amount.
2. Relative intensity or amount, as of a quality or attribute: degree, extent, magnitude, proportion. See big/small/amount.
3. A means by which individuals are compared and judged: benchmark, criterion, gauge, mark, standard, test, touchstone, yardstick. See usual/unusual.
4. The act or process of ascertaining dimensions, quantity, or capacity: measurement, mensuration, metrology. See big/small/amount.
5. That which is allotted: allocation, allotment, allowance, dole, lot, part, portion, quantum, quota, ration, share, split. Informal cut. Slang divvy. See collect/distribute.
6. Avoidance of extremes of opinion, feeling, or personal conduct: moderateness, moderation, temperance. See edge/center.
7. An action calculated to achieve an end. maneuver, move, procedure, step, tactic. See action/inaction.
8. The formal product of a legislative or judicial body: act, assize, bill¹, enactment, law, legislation, lex, statute. See law.
9. The patterned, recurring alternation of contrasting elements, such as stressed and unstressed notes in music: beat, cadence, cadency, meter, rhythm, swing. See repetition.

verb

1. To ascertain the dimensions, quantity, or capacity of: gauge. Archaic mete. Idioms: take the measure of. See big/small/amount.
2. To fix the limits of: bound², delimit, delimitate, demarcate, determine, limit, mark (off or out). See limited/unlimited.

phrasal verb - measure out

To set aside or distribute as a share: admeasure, allocate, allot, allow, apportion, assign, give, lot, mete (out). See collect/distribute.

phrasal verb - measure up

To be equal or alike: compare, correspond, equal, match, parallel, touch. Informal stack up. See same/different/compare.

In addition to the idiom beginning with measure, also see beyond measure; for good measure; in some measure; made to measure; take someone's measure.

Definition: preventive or institutive action
Antonyms: ignorance, inaction

v

Definition: calculate, judge
Antonyms: estimate, guess

(1) English term ofc 1550-1650 for a sequence of dance steps in slow or moderate duple time roughly corresponding to one strain of music. Measures were usually set to pavans and almans.

(2) American term, equivalent to the English ‘bar’, for the metrical units marked off along the staff by vertical lines (bars or bar-lines). See Bar.

measure, an older word for metre. The term is also used to refer to any metrical unit such as a foot, a dipody, or a line.

Poetic rhythm or cadence as determined by the syllables in a line of poetry with respect to quantity and accent; also, meter; also, a metrical foot.

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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.

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. 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 Rⁿ, n=1,2,3,.... For instance, the Lebesgue measure of [0,1] in the real numbers is its length in the non-formal sense of the word, specifically 1.

To qualify as a measure (see Definition below), a function that assigns a non-negative real number or infinity to a set's subsets must satisfy a few conditions. One important condition is countable additivity. This condition states that the size of the union of a sequence of disjoint subsets is equal to the sum of the sizes of the subsets. However, it is in general impossible to consistently associate a size to each subset of a given set and also satisfy the other axioms of a measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the subsets on which the measure is to be defined are called measurable and they are required to form a sigma-algebra, meaning that unions, intersections and complements of sequences of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be consistently defined, are necessarily complex to the point of incomprehensibility, in a sense badly mixed up with their complement; 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 century by Emile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure.

Contents 1 Definition 2 Properties 2.1 Monotonicity 2.2 Measures of infinite unions of measurable sets 2.3 Measures of infinite intersections of measurable sets 3 Sigma-finite measures 4 Completeness 5 Examples 6 Non-measurable sets 7 Generalizations 8 See also 9 References 10 External links

Definition

Let Σ be a σ-algebra over a set X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:

• Non-negativity:

μ(E) ≥ 0 for all E∊Σ.

• Null empty set:

μ(Ø) = 0.

• Countable additivity (or σ-additivity): For all countable collections {E_i} of pairwise disjoint sets in Σ:

The second condition may be treated as a special case of countable additivity, if the empty collection is allowed as a countable collection (and the empty sum is interpreted as 0). Otherwise, if the empty collection is disallowed (but finite collections are allowed), the second condition still follows from countable additivity provided, however, that there is at least one set having finite measure.

The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets, and the triple (X, Σ, μ) is called a measure space.

If only the second and third conditions are met, and μ takes on at most one of the values ±∞, then μ is called a signed measure.

A probability measure is a measure with total measure one (i.e., μ(X) = 1); a probability space is a measure space with a probability measure.

For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other authors. For more details see Radon measure.

Properties

Several further properties can be derived from the definition of a countably additive measure.

Monotonicity

A measure μ is monotonic: If E₁ and E₂ are measurable sets with E₁ ⊆ E₂ then

Measures of infinite unions of measurable sets

A measure μ is countably subadditive: If E₁, E₂, E₃, … is a countable sequence of sets in Σ, not necessarily disjoint, then

A measure μ is continuous from below: If E₁, E₂, E₃, … are measurable sets and E_n is a subset of E_{n + 1} for all n, then the union of the sets E_n is measurable, and

Measures of infinite intersections of measurable sets

A measure μ is continuous from above: If E₁, E₂, E₃, … are measurable sets and E_{n + 1} is a subset of E_n for all n, then the intersection of the sets E_n is measurable; furthermore, if at least one of the E_n has finite measure, then

This property is false without the assumption that at least one of the E_n has finite measure. For instance, for each n ∈ N, let

which all have infinite Lebesgue measure, but the intersection is empty.

Sigma-finite measures

Main article: Sigma-finite measure

A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). It is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a countable union of sets with finite measure.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k,k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

Completeness

A measurable set X is called a null set if μ(X)=0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.

A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).

Examples

Some important measures are listed here.

• The counting measure is defined by μ(S) = number of elements in S.
• The Lebesgue measure on R is a complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0,1]) = 1; and every other measure with these properties extends Lebesgue measure.
• Circular angle measure is invariant under rotation.
• The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
• The Hausdorff measure which is a refinement of the Lebesgue measure to some fractal sets.
• Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0,1]). Such a measure is called a probability measure. See probability axioms.
• The Dirac measure δ_a (cf. Dirac delta function) is given by δ_a(S) = χ_S(a), where χ_S is the characteristic function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure.

Non-measurable sets

Main article: Non-measurable set

If the axiom of choice is assumed to be true, not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.

Generalizations

For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used mainly in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures.

Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L^∞ and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice.

A charge is a generalization in both directions: it is a finitely additive, signed measure.

The remarkable result in integral geometry known as Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compact convex sets in Rⁿ consists (up to scalar multiples) of one "measure" that is "homogeneous of degree k" for each k = 0, 1, 2, ..., n, and linear combinations of those "measures". "Homogeneous of degree k" means that rescaling any set by any factor c > 0 multiplies the set's "measure" by c^k. The one that is homogeneous of degree n is the ordinary n-dimensional volume. The one that is homogeneous of degree n − 1 is the "surface volume". The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer. The one that is homogeneous of degree 0 is the Euler characteristic.

See also

Look up measurable in Wiktionary, the free dictionary.

• Outer measure
• Inner measure
• Hausdorff measure
• Product measure
• Pushforward measure
• Lebesgue measure
• Vector measure
• Almost everywhere
• Lebesgue integration
• Caratheodory extension theorem
• Measurable function
• Geometric measure theory
• Volume form
• Fuzzy measure theory

References

• R. G. Bartle, 1995. The Elements of Integration and Lebesgue Measure. Wiley Interscience.
• Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1  Chapter III.
• R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press.
• Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0-471-317160-0  Second edition.
• D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.
• Paul Halmos, 1950. Measure theory. Van Nostrand and Co.
• R. Duncan Luce and Louis Narens (1987). "measurement, theory of," The New Palgrave: A Dictionary of Economics, v. 3, pp. 428-32.
• M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.
• K. P. S. Bhaskara Rao and M. Bhaskara Rao (1983). Theory of Charges: A Study of Finitely Additive Measures. London: Academic Press. pp. x + 315. ISBN 0-1209-5780-9. 
• Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.

External links

• Measure theory for dummies, pdf article

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Dansk (Danish)
v. tr. - måle, opmåle, bedømme, anslå
v. intr. - have mål, tage mål, gøre det muligt at tage mål
n. - mål, kvantum, grad, udstrækning, omfang

idioms:

• beyond measure    over al måde
• in some measure    i nogen grad
• measure off    udmåle
• measure out    udmåle, uddele
• measure up    tage mål, være egnet, kvalificeret
• measure up to    stå mål med, passe med
• take measures    tage forholdsregler

Nederlands (Dutch)
meten, afmeten, opmeten, uitmeten, taxeren, reguleren, bepaalde maat hebben, toedienen, reizen, maatregel, maat, versvoet, beschikking, wetsvoorstel, dans, metrum, pagina-/ kolombreedte, gematigdheid, portie, grens, afmeting, schatting, (gemeten) hoeveelheid, meetinstrument, matenstelsel, melodie, deler (wiskunde)

Français (French)
v. tr. - mesurer, comparer qch à (des efforts)
v. intr. - mesurer (qn, qch)
n. - unité de mesure, mesures, sur mesure (un vêtement), instrument de mesure, (fig) certain, mesure (de l'augmentation des prix), indication, critère, énorme, mesure (contre) (pour faire), (Pol, Jur) mesure, (Mus, Littérat, Danse) mesure

idioms:

• beyond measure    énormément, extrêmement
• get someone's measure    savoir ce qu'une personne vaut
• have the measure of    prendre la mesure de qn, jauger qn
• in a measure    dans une certaine mesure
• in some measure    dans une certaine mesure
• measure off    mesurer
• measure out    mesurer, doser, répartir (qch)
• measure up    mesurer (qch), faire le poids, être à la mesure de
• measure up to    être à la hauteur de, soutenir la comparaison avec
• take measures    prendre des mesures
• take the measure of    jauger (qn), prendre la mesure de

Deutsch (German)
n. - Maßnahme, Maß, Maßstab, Versmaß, Takt
v. - messen, ausmessen, Maß nehmen, abschätzen

idioms:

• beyond measure    maßlos
• get someone's measure    jmdn. durchschauen
• have the measure of    jmdn. durchschauen
• in a measure    in gewisser Hinsicht, gewissermaßen
• in some measure    in gewisser Hinsicht
• measure off    ausmessen
• measure out    abmessen
• measure up    abmessen
• measure up to    entsprechen, (Anforderungen) gewachsen sein
• take measures    Maßnahmen ergreifen
• take the measure of    etw. ausmessen

Ελληνική (Greek)
n. - μέτρο, μέγεθος, ποσότητα, μεζούρα, ενέργεια, (μαθημ.) διαιρέτης
v. - μετρώ, δοκιμάζω, είναι, έχει διαστάσεις

idioms:

• beyond measure    υπέρμετρα, απεριόριστος
• in some measure    εν μέρει, μέχρις ενός ορίου/σημείου
• measure off    καταμετρώ, καθορίζω τα όρια
• measure out    επιμερίζω, διανέμω σε μετρημένες ποσότητες
• measure up    μετρώ, παίρνω μέτρα, έχω ό, τι χρειάζεται
• measure up to    είμαι του ιδίου επιπέδου με, είμαι ισάξιος τού
• take measures    λαμβάνω μέτρα

Italiano (Italian)
misurare, misura, metro, moderazione

idioms:

• beyond measure    oltre ogni misura
• in some measure    in qualche misura
• measure off    marcare i limiti
• measure out    dosare
• measure up    essere all'altezza
• measure up to    misurarsi con
• take measures    prendere dei provvedimenti

Português (Portuguese)
n. - medida (f), quantidade (f), camada (f) (Geol.)
v. - medir

idioms:

• beyond measure    sem limite
• in some measure    até certo ponto
• measure off    cortar
• measure out    medir precisamente
• measure up    estar à altura
• measure up to    ter gabarito
• take measures    tirar medidas

Русский (Russian)
мера, размер, степень, мероприятие, измерять, сравнивать, соразмерять, снимать мерку, иметь размеры

idioms:

• beyond measure    чрезмерно
• in some measure    до известной степени
• measure off    отмерять
• measure out    распределять
• measure up    соответствовать требованиям, достигать уровня
• measure up to    быть достойным кого-либо
• take measures    принимать меры

Español (Spanish)
v. tr. - medir, tomar las medidas, mensurar, aforar, ajustar, recorrer
v. intr. - medir, tomar las medidas
n. - regla, norma, medida, tamaño, dimensiones, magnitud, contador, medidor, moderación

idioms:

• beyond measure    excesivamente, desmedidamente
• get someone's measure    obtener las medidas de alguien
• have the measure of    tener calado a alguien/algo, agarrarle la onda
• in a measure    en la medida que
• in some measure    hasta cierto punto
• measure off    medir
• measure out    pesar, medir, repartir, distribuir, dar una porción de
• measure up    tomar las medidas, dar la talla, estar a la altura de las circunstancias, valorar, juzgar
• measure up to    estar a la altura de algo
• take measures    tomar medidas
• take the measure of    tomar la medida de

Svenska (Swedish)
n. - mått, storlek, måttredskap, mån, grad, gräns, åtgärd, lagförslag, satsbredd, versmått, takt, visa, dans, avstånd, skikt, divisor
v. - mäta, beräkna, bedöma, avpassa, tillryggalägga

中文（简体）(Chinese (Simplified))
测量, 估量, 测度, 量, 尺寸, 量度标准, 量度器

idioms:

• beyond measure    无可估量, 极度
• in some measure    多少, 稍稍多少, 稍稍
• measure off    划分
• measure out    按量配给
• measure up    合格, 符合标准
• measure up to    符合, 达到
• take measures    设法, 着手

中文（繁體）(Chinese (Traditional))
v. tr. - 測量, 估量, 測度
v. intr. - 量
n. - 尺寸, 量度標準, 量度器

idioms:

• beyond measure    無可估量, 極度
• in some measure    多少, 稍稍多少, 稍稍
• measure off    劃分
• measure out    按量配給
• measure up    合格, 符合標準
• measure up to    符合, 達到
• take measures    設法, 著手

한국어 (Korean)
v. tr. - 측정하다, 판단하다, 조화롭게 하다, 자세히 보다
v. intr. - 치수를 측정하다, ~의 치수이다
n. - 측정, 단위, 액수, 기준, 적정량

idioms:

• beyond measure    지나치게
• in some measure    다소
• measure off    재어서 끊다, 구획을 정리하다
• measure out    재어서 분배하다
• measure up    ~의 치수를 재다, 가늠하다, 능력이 있다
• measure up to    치수가 ~에 달하다, 적당하다, 일치하다
• take measures    치수를 재다

日本語 (Japanese)
n. - 基準, 計量法, 測定, 測量, 寸法, 程度, 適度, 限度, 度量測定器具, 韻律, 小節, 処置, 法案, 測度, 大きさ, 測量法
v. - 計る, 測定単位である, 比較する, 測る, 判断する, 考量する, 釣り合わせる, 測定できる, …だけの長さがある, 見積もる

idioms:

• beyond measure    過度に, 無際限に
• measure off    測って切る, 区別する
• measure out    量り分ける, 分配する
• measure up    期待にそう, 標準に達する
• measure up to    …にかなう
• take measures    寸法をはかる

العربيه (Arabic)
‏(الاسم) مقاس, , ميزان تزميني, بحر, وزن تفعيلي, طريقه, اجراء (فعل) يقيس‏

עברית (Hebrew)
v. tr. - ‮מדד, אמד‬
v. intr. - ‮היה אורכו, היה רוחבו, היה גודלו‬
n. - ‮מידה, שיעור, כלי מדידה, אמצעי, צעד, חוק, משקל, קצב, תיבה (במוסיקה)‬

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