measure

—measurer, n.

/mezh"euhr/, n., v., measured, measuring.

n.

1. a unit or standard of measurement: weights and measures.

2. a system of measurement: liquid measure.

3. an instrument, as a graduated rod or a container of standard capacity, for measuring.

4. the extent, dimensions, quantity, etc., of something, ascertained esp. by comparison with a standard: to take the measure of a thing.

5. the act or process of ascertaining the extent, dimensions, or quantity of something; measurement.

6. a definite or known quantity measured out: to drink a measure of wine.

7. any standard of comparison, estimation, or judgment.

8. a quantity, degree, or proportion: in large measure.

9. a moderate amount: to live with a measure of enjoyment.

10. a limit, or an extent or degree not to be exceeded: to know no measure.

11. reasonable bounds or limits: to know no measure.

12. a legislative bill or enactment: The senate passed the new measure.

13. Usually, measures. actions or procedures intended as a means to an end: to take measures to avert suspicion.

14. a short rhythmical movement or arrangement, as in poetry or music.

15. a particular kind of such arrangement.

16. the music contained between two bar lines; bar.

17. a metrical unit.

18. an air or melody.

19. a slow, dignified dance.

20. Print. the width, measured in ems or picas, to which a column or page of printed matter is set.

21. measures, Geol. beds; strata.

22. Math. an abstraction of the property of length; a set function assigning to each set of a collection of sets a value, usu. having the properties of sigma finiteness and fnite additivity, the functional value of the whole collection being greater than zero.

23. beyond measure, too much to be reckoned; immeasurably; extremely: The suffering that they endured was beyond measure.

24. for good measure, as an extra: In addition to dessert, they served chocolates for good measure.

25. have or take someone's measure, to judge or assess someone's character, capabilities, etc.; size up: During their conversation she was taking his measure as a prospective employee.

26. in a or some measure, to some extent or degree: His conclusion is justified in some measure.

v.t.

27. to ascertain the extent, dimensions, quantity, capacity, etc., of, esp. by comparison with a standard: to measure boundaries.

28. to mark off or deal out by way of measurement (often fol. by off or out): to measure out two cups of flour.

29. to estimate the relative amount, value, etc., of, by comparison with some standard: to measure the importance of an issue.

30. to judge or appraise by comparison with something or someone else: to measure Corneille against Racine.

31. to serve as the measure of: Her sacrifices measure the degree of her love.

32. to adjust or proportion: to measure a portion to one's liking.

33. to bring into comparison or competition: to measure one's strength with another's.

34. to travel over; traverse: to measure a room with great strides.

v.i.

35. to take measurements.

36. to admit of measurement.

37. to be of a specified measure.

38. measure one's length, to fall or be knocked down; fall flat: He missed a step in the dark and measured his length at the bottom.

39. measure swords,

a. to test one's preparedness for a contest or encounter.

b. to battle with swords.

c. to fight, compete, etc.: The producer of the poorly reviewed show decided to measure swords with the critics.

40. measure up,

a. to reach a certain standard: The exhibition didn't measure up to last year's.

b. to be capable or qualified: As an administrator, he couldn't quite measure up.

[1250-1300; ME mesure < MF < L mensura equiv. to mens(us) (ptp. of metiri to measure, mete) + -ura -URE]

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▪ mathematics

      in mathematics, generalization of the concepts of length and area to arbitrary sets of points not composed of intervals or rectangles. Abstractly, a measure is any rule for associating with a set a number that retains the ordinary measurement properties of always being nonnegative and such that the sum of the parts equals the whole. More formally, the measure of the union of two nonoverlapping sets is equal to the sum of their individual measures. The measure of an elementary set composed of a finite number of nonoverlapping rectangles can be defined simply as the sum of their areas found in the usual manner. (And analogously, the measure of a finite union of nonoverlapping intervals is the sum of their lengths.)

      For other sets, such as curved regions or vaporous regions with missing points, the concepts of outer and inner measure must first be defined. The outer measure of a set is the number that is the lower bound of the area of all elementary rectangular sets containing the given set, while the inner measure of a set is the upper bound of the areas of all such sets contained in the region. If the inner and outer measures of a set are equal, this number is called its Jordan measure, and the set is said to be Jordan measurable.

      Unfortunately, many important sets are not Jordan measurable. For example, the set of rational numbers from zero to one is not composed of a finite number of intervals, and so no length is defined for it. It has a measure, however, that can be found in the following way: The rational numbers are countable (can be put in a one-to-one relationship with the counting numbers 1, 2, 3,…), and each successive number can be covered by intervals of length 1/8, 1/16, 1/32,…, the total sum of which is 1/4, calculated as the sum of the infinite geometric series (infinite series). The rational numbers could also be covered by intervals of lengths 1/16, 1/32, 1/64,…, the total sum of which is 1/8. By starting with smaller and smaller intervals, the total length of intervals covering the rationals can be reduced to smaller and smaller values that approach the lower bound of zero, and so the outer measure is 0. The inner measure is always less than or equal to the outer measure, so it must also be 0. Therefore, although the set of rational numbers is infinite, their measure is 0. In contrast, the irrational numbers (irrational number) from zero to one have a measure equal to 1; hence, the measure of the irrational numbers is equal to the measure of the real numbers (real number)—in other words, “almost all” real numbers are irrational numbers. The concept of measure based on countably infinite collections of rectangles is called Lebesgue measure.

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