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Measure (mathematics)

From Wikipedia, the free encyclopedia

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.
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 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 equal to 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 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, the 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. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

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Transcription

I have two seemingly unrelated challenges for you. The first relates to music, and the second gives a foundational result in measure theory, which is the formal underpinning for how mathematicians define integration and probability. The second challenge, which I’ll get to about halfway through the video, has to do with covering numbers with open sets, and is very counter-intuitive. Or at least, when I first saw it I was confused for a while. Foremost, I’d like to explain what’s going on, but I also plan to share a surprising connection it has with music. Here’s the first challenge. I’m going to play a musical note with a given frequency, let’s say 220 hertz, then I’m going to choose some number between 1 and 2, which we’ll call r, and play a second musical note whose frequency is r times the frequency of the first note, 220. For some values of this ratio r, like 1.5, the two notes will sound harmonious together, but for others, like the square root of 2, they sound cacophonous. Your task it to determine whether a given ratio r will give a pleasant sound or an unpleasant one just by analyzing the number and without listening to the notes. One way to answer, especially if your name is Pythagoras, might be that two notes sound good when the ratio is a rational number, and bad when it is irrational. For instance, a ratio of 3/2 gives a musical fifth, 4/3 gives a musical fourth, of 8/5 gives a minor sixth, etc. Here’s my best guess for why this is the case: a musical note is made up of beats played in rapid succession, for instance 220 beats per second. When the ratio of frequencies of two notes is rational, there is a detectable pattern in those beats, which, when we slow it down, we hear as a rhythm instead of as a harmony. Evidently when our brains pick up on this pattern, two notes sound nice together. However, most rational numbers actually sound pretty bad, like 211/198, or 1093/826. The issue, of course, is that these rational number are somehow more “complicated” than the other ones, our ears don’t pick up on the pattern of the beats. One simple way to measure the complexity of a rational number is to consider the size of its denominator when it is written in reduced form. So we might edit our original answer to only admit fractions with low denominators, say less than 10. Even still, this doesn’t quite capture harmoniousness, since plenty of notes sound good together even when the ratio of their frequencies is irrational, so long as it is close to a harmonious rational number. And it’s a good thing, too, because many instruments such as pianos are not tuned in terms of rational intervals, but are tuned such that each half-step increase corresponds with multiplying the original frequency by the 12th root of 2, which is irrational. If you’re curious about why this is done, Henry at minutephysics recently did a video which gives a very nice explanation. This means that if you take a harmonious interval, like a fifth, the ratio of frequencies when played on a piano will not be a nice rational number like you expect, in this case 3/2, but will instead be some power of the 12th root of 2, in this case 2^{7/12}, which is irrational, but very close to 3/2. Similarly, a musical fourth corresponds to 2^{5/12}, which is very close to 4/3. In fact, the reason it works so well to have 12 notes in the chromatic scale is that powers of the 12th root of 2 have a strange tendency to be within a 1% margin of error of simple rational numbers. So now you might say a ratio r will produce a harmonious pair of notes if it is sufficiently close to a rational number with a sufficiently small denominator. How close depends on how discerning your ear is, and how small a denominator depends on the intricacy of harmonic patterns your ear has been trained to pick up on. After all, maybe someone with a particularly acute musical sense would be able to hear and find pleasure in the pattern resulting from more complicated fractions like 23/21 or 35/43, as well as numbers closely approximating these fractions. This leads to an interesting question: Suppose there is a musical savant, who find pleasure in all pairs of notes whose frequencies have a rational ratio, even super complicated ratios that you and I would find cacophonous. Is it the case that she would find all ratios r between 1 and 2 harmonious, even the irrational ones? After all, for any given real number you can always find rational numbers arbitrarily close it, just as 3/2 is close to 2^{7/12}. Well, this brings us to challenge number 2. Mathematicians like to ask riddles about covering various sets with open intervals, and the answers to these riddles have a strange tendency to become famous lemmas and theorems. By “open interval”, I just mean the continuous stretch of real numbers strictly greater than some number a, but strictly less than some other number b, where b is of course greater than a. My challenge to you involves covering all the rational numbers between 0 and 1 with open intervals. When I say “cover”, all that means is that each particular rational number lies in at least one of your intervals. The most obvious way to do this is to just use the entire interval from 0 to 1 itself and call it done, but the challenge here is that the sum of the lengths of your intervals must be strictly less than 1. To aid you in this seemingly impossible task, you are allowed to use infinitely many intervals. Even still, the task might feel impossible, since the rational numbers are dense in the real numbers, meaning any stretch, no matter how small, contains infinitely many rational numbers. So how could you possibly cover all rational numbers without just covering the entire interval from 0 to 1 itself, which would mean the total length of your open intervals has to be at least the length of the entire interval from 0 to 1. Then again, I wouldn’t be talking about this if there was not a way to do it. First, we enumerate the rational numbers between 0 and 1, meaning we organize them into an infinitely long list. There are many ways to do this, but one natural way I’ll choose is start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don’t write down 2/4 since it has already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. Every fraction will appear exactly once in this list, in its reduced form, and it gives us a meaningful way to talk about the “first” rational number, the “second” rational number, the 42nd rational number, things like that. Next, to ensure that each rational is covered, we are going to assign one specific interval to each rational. Once we remove the intervals from the geometry of our setup and just think of them in a list, each one responsible for only one rational number, it seems much clearer that the sum of their lengths can be less than 1, since each particular interval can be as small as you want and still cover its designated rational. In fact, the sum can be any positive number. Just choose an infinite sum with positive terms that converges to 1, like ½+¼+⅛+... on and on with powers of 2, then choose any desired value epsilon>0, like 0.5, and multiply all terms by epsilon so that we have an infinite sum converging to epsilon. Now scale the nth interval to have a length equal to the nth term in the sum. Notice, this means your intervals start getting really small, really fast, so small that you can’t really see most of them in this animation, but it doesn’t matter, since each one is only responsible for covering one rational. I’ve said it already, by I’ll say it again because it’s so amazing: epsilon can be whatever positive number we want, so not only can our sum be less than 1, it can be arbitrarily small! This is one of those results where even after seeing the proof, it still defies intuition. The discord here is that the proof has us thinking analytically, with the rational numbers in a list, but our intuition has us thinking geometrically, with the rationals as a dense set on the interval, where you can’t skip over any continuous stretch of numbers since each stretch contains infinitely many rationals. So let’s get a visual understanding of what’s going on. Brief side note here: I had trouble deciding on how to illustrate small open intervals, since if I scale the parentheses with the interval, you won’t be able to see them at all, but if I just push the parentheses together, they cross over in a way that it potentially confusing. Nevertheless, I decided to go with the ugly chromosomal cross, so keep in mind that the interval they represent is the tiny stretch between the centers of each parenthesis. Okay, back to the visual intuition. Consider when epsilon = 0.3, meaning if I choose a number between 0 and 1 at random, there is a 70% that it is outside all those infinitely many intervals. What does it look like to be outside the intervals? Well, the square root of 2 over 2 is among those 70%, and I’m going to zoom in it. As I do so I’ll draw the first 10 intervals in the list within our scope of vision. As we get closer to the square root of 2 over 2, even though you will always find rationals within your field of view, the intervals placed on top of those rationals get really small really fast. One might say that for any sequence of rational numbers approaching the square root of 2 over 2, the intervals covering the elements of this sequence shrink faster than that sequence converges. Notice, intervals are really small if they show up very late in the list, and rationals show up late in the list when they have large denominators, so the fact that the square root of 2 over 2 is among the 70% not covered by our intervals is in a sense a way to formalize the otherwise vague idea that the only rational numbers “close” to it have large denominators. That is to say, the square root of 2 over 2 is cacophonous. In fact, let’s use a smaller epsilon, say 0.01, and shift our setup to lie on top of the interval from 1 to 2 instead of from 0 to 1. Then which numbers fall among the elite 1% covered by our tiny intervals? Almost all of them are harmonious! For instance, the harmonious irrational number 2^{7/12} is very close to 3/2, which has a relatively fat interval sitting on top of it, and the interval around 4/3 is smaller, but still fat enough to cover 2^{5/12}. Which members of the 1% are cacophonous? Well, the cacophonous rationals, meaning those with high denominators, and irrationals that are very very very close to them. However, think of the savant who finds harmonic patterns in all rational numbers. You could imagine that for her, harmonious numbers are precisely those 1% covered by the intervals, provided that her tolerance for error goes down exponentially for more complicated rationals. In other words, the seemingly paradoxical fact that you can have a collection of intervals densely populate a range while only covering 1% of its values corresponds to the fact that harmonious numbers are rare, even for the savant. I’m not saying this makes it the result more intuitive, in fact, I find it quite surprising that the savant I defined could find 99% of all ratios cacophonous, but the fact that these two ideas are connected was simply too beautiful not to share.

Contents

Definition

Countable additivity of a measure μ: The measure of a countable disjoint union is the same as the sum of all measures of each subset.
Countable additivity of a measure μ: The measure of a countable disjoint union is the same as the sum of all measures of each subset.

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

  • Non-negativity: For all E in Σ: μ(E) ≥ 0.
  • Null empty set: .
  • Countable additivity (or σ-additivity): For all countable collections of pairwise disjoint sets in Σ:

One may require that at least one set E has finite measure. Then the empty set automatically has measure zero because of countable additivity, because

which implies (since the sum on the right thus converges to a finite value) that .

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

The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets. If and are two measurable spaces, then a function is called measurable if for every Y-measurable set , the inverse image is X-measurable – i.e.: . In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. See also Measurable function#Term usage variations about another setup.

A triple (X, Σ, μ) is called a measure space. 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 sources. For more details, see the article on Radon measures.

Examples

Some important measures are listed here.

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

In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.

Basic properties

Let μ be a measure.

Monotonicity

If E1 and E2 are measurable sets with E1 ⊆ E2 then

Measure of countable unions and intersections

Subadditivity

For any countable sequence E1, E2, E3, ... of (not necessarily disjoint) measurable sets En in Σ:

Continuity from below

If E1, E2, E3, ... are measurable sets and En is a subset of En + 1 for all n, then the union of the sets En is measurable, and

Continuity from above

If E1, E2, E3, ... are measurable sets and for all n, En + 1En, then the intersection of the sets En is measurable; furthermore, if at least one of the En has finite measure, then

This property is false without the assumption that at least one of the En has finite measure. For instance, for each nN, let En = [n, ∞) ⊂ R, which all have infinite Lebesgue measure, but the intersection is empty.

Sigma-finite measures

A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure μ is proportional to the probability measure . A measure μ is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a 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'.

s-finite measures

A measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.

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).

Additivity

Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set and any set of nonnegative define:

That is, we define the sum of the to be the supremum of all the sums of finitely many of them.

A measure on is -additive if for any and any family of disjoint sets the following hold:

Note that the second condition is equivalent to the statement that the ideal of null sets is -complete.

Non-measurable sets

If the axiom of choice is assumed to be true, it can be proved that 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.[2] 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 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, also known as a content. 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. 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. Contents remain useful in certain technical problems in geometric measure theory; this is the theory of Banach measures.

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

See also

References

  1. ^ Halmos, Paul (1950), Measure theory, Van Nostrand and Co.
  2. ^ Rao, M. M. (2012), Random and Vector Measures, Series on Multivariate Analysis, 9, World Scientific, ISBN 978-981-4350-81-5, MR 2840012.

Bibliography

External links

This page was last edited on 5 March 2019, at 05:05
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