Measure (mathematics)

Summary

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.

Informally, a measure has the property of being monotone in the sense that if is a subset of the measure of is less than or equal to the measure of Furthermore, the measure of the empty set is required to be 0.

The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.

DefinitionEdit

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

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

  • Non-negativity: For all   in   we have  
  • Null empty set:  
  • Countable additivity (or  -additivity): For all countable collections   of pairwise disjoint sets in Σ,
     

If at least one set   has finite measure, then the requirement that   is met automatically. Indeed, by countable additivity,

 
and therefore  

If the condition of non-negativity is omitted but the second and third of these conditions are met, and   takes on at most one of the values   then   is called a signed measure.

The pair   is called a measurable space, and the members of   are called measurable sets.

A triple   is called a measure space. A probability measure is a measure with total measure one – that is,   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 topological vector 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.

InstancesEdit

Some important measures are listed here.

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic 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 for example, 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.

  • Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.
  • Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.

Basic propertiesEdit

Let   be a measure.

MonotonicityEdit

If   and   are measurable sets with   then

 

Measure of countable unions and intersectionsEdit

SubadditivityEdit

For any countable sequence   of (not necessarily disjoint) measurable sets   in  

 

Continuity from belowEdit

If   are measurable sets that are increasing (meaning that  ) then the union of the sets   is measurable and

 

Continuity from aboveEdit

If   are measurable sets that are decreasing (meaning that  ) then the intersection of the sets   is measurable; furthermore, if at least one of the   has finite measure then

 

This property is false without the assumption that at least one of the   has finite measure. For instance, for each   let   which all have infinite Lebesgue measure, but the intersection is empty.

Other propertiesEdit

CompletenessEdit

A measurable set   is called a null set if   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   which differ by a negligible set from a measurable set   that is, such that the symmetric difference of   and   is contained in a null set. One defines   to equal  

μ{x : f(x) ≥ t} = μ{x : f(x) > t} (a.e.)Edit

If   is  -measurable, then

 
for almost all  [1] This property is used in connection with Lebesgue integral.
Proof

Both   and   are monotonically non-increasing functions of   so both of them have at most countably many discontinuities and thus they are continuous almost everywhere, relative to the Lebesgue measure. If   then   so that   as desired.

If   is such that   then monotonicity implies

 
so that   as required. If   for all   then we are done, so assume otherwise. Then there is a unique   such that   is infinite to the left of   (which can only happen when  ) and finite to the right. Arguing as above,   when   Similarly, if   and   then  

For   let   be a monotonically non-decreasing sequence converging to   The monotonically non-increasing sequence   of members of   has at least one finitely  -measurable component, and

 
Continuity from above guarantees that
 
The right-hand side   then equals   if   is a point of continuity of   Since   is continuous almost everywhere, this completes the proof.

AdditivityEdit

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.

Sigma-finite measuresEdit

A measure space   is called finite if   is a finite real number (rather than  ). Nonzero finite measures are analogous to probability measures in the sense that any finite measure   is proportional to the probability measure   A measure   is called σ-finite if   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   for all integers   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.[original research?] They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

Strictly localizable measuresEdit

Semifinite measuresEdit

Let   be a set, let   be a sigma-algebra on   and let   be a measure on   We say   is semifinite to mean that for all    [2]

Semifinite measures generalize sigma-finite measures, in such a way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.)

Basic examplesEdit

  • Every sigma-finite measure is semifinite.
  • Assume   let   and assume   for all  
    • We have that   is sigma-finite if and only if   for all   and   is countable. We have that   is semifinite if and only if   for all  [3]
    • Taking   above (so that   is counting measure on  ), we see that counting measure on   is
      • sigma-finite if and only if   is countable; and
      • semifinite (without regard to whether   is countable). (Thus, counting measure, on the power set   of an arbitrary uncountable set   gives an example of a semifinite measure that is not sigma-finite.)
  • Let   be a complete, separable metric on   let   be the Borel sigma-algebra induced by   and let   Then the Hausdorff measure   is semifinite.[4]
  • Let   be a complete, separable metric on   let   be the Borel sigma-algebra induced by   and let   Then the packing measure   is semifinite.[5]

Involved exampleEdit

The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to   It can be shown there is a greatest measure with these two properties:

Theorem (semifinite part)[6] — For any measure   on   there exists, among semifinite measures on   that are less than or equal to   a greatest element  

We say the semifinite part of   to mean the semifinite measure   defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:

  •  [6]
  •  [7]
  •  [8]

Since   is semifinite, it follows that if   then   is semifinite. It is also evident that if   is semifinite then  

Non-examplesEdit

Every   measure that is not the zero measure is not semifinite. (Here, we say   measure to mean a measure whose range lies in  :  ) Below we give examples of   measures that are not zero measures.

  • Let   be nonempty, let   be a  -algebra on   let   be not the zero function, and let   It can be shown that   is a measure.
    •  [9]
      •      [10]
  • Let   be uncountable, let   be a  -algebra on   let   be the countable elements of   and let   It can be shown that   is a measure.[2]

Involved non-exampleEdit

Measures that are not semifinite are very wild when restricted to certain sets.[Note 1] Every measure is, in a sense, semifinite once its   part (the wild part) is taken away.

— A. Mukherjea and K. Pothoven, Real and Functional Analysis, Part A: Real Analysis (1985)

Theorem (Luther decomposition)[11][12] — For any measure   on   there exists a   measure   on   such that   for some semifinite measure   on   In fact, among such measures   there exists a least measure   Also, we have  

We say the   part of   to mean the measure   defined in the above theorem. Here is an explicit formula for  :  

Results regarding semifinite measuresEdit

  • Let   be   or   and let   Then   is semifinite if and only if   is injective.[13][14] (This result has import in the study of the dual space of  .)
  • Let   be   or   and let   be the topology of convergence in measure on   Then   is semifinite if and only if   is Hausdorff.[15][16]
  • (Johnson) Let   be a set, let   be a sigma-algebra on   let   be a measure on   let   be a set, let   be a sigma-algebra on   and let   be a measure on   If   are both not a   measure, then both   and   are semifinite if and only if    for all   and   (Here,   is the measure defined in Theorem 39.1 in Berberian '65.[17])

Localizable measuresEdit

Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures.

Let   be a set, let   be a sigma-algebra on   and let   be a measure on  

  • Let   be   or   and let   Then   is localizable if and only if   is bijective (if and only if   "is"  ).[18][14]

s-finite measuresEdit

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.

Non-measurable setsEdit

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.

GeneralizationsEdit

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. Observe, however, that complex measure is necessarily of finite variation, hence complex measures include finite signed measures but not, for example, the Lebesgue measure.

Measures that take values in Banach spaces have been studied extensively.[19] 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   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.[20] (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range its a bounded subset of R.)

See alsoEdit

NotesEdit

  1. ^ One way to rephrase our definition is that   is semifinite if and only if   Negating this rephrasing, we find that   is not semifinite if and only if   For every such set   the subspace measure induced by the subspace sigma-algebra induced by   i.e. the restriction of   to said subspace sigma-algebra, is a   measure that is not the zero measure.

BibliographyEdit

  • Robert G. Bartle (1995) The Elements of Integration and Lebesgue Measure, Wiley Interscience.
  • Bauer, H. (2001), Measure and Integration Theory, Berlin: de Gruyter, ISBN 978-3110167191
  • Bear, H.S. (2001), A Primer of Lebesgue Integration, San Diego: Academic Press, ISBN 978-0120839711
  • Berberian, Sterling K (1965). Measure and Integration. MacMillan.
  • Bogachev, V. I. (2006), Measure theory, Berlin: Springer, ISBN 978-3540345138
  • 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.
  • Edgar, Gerald A (1998). Integral, Probability, and Fractal Measures. Springer. ISBN 978-1-4419-3112-2.
  • Folland, Gerald B (1999). Real Analysis: Modern Techniques and Their Applications (Second ed.). Wiley. ISBN 0-471-31716-0.
  • Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp.
  • Fremlin, D.H. (2016). Measure Theory, Volume 2: Broad Foundations (Hardback ed.). Torres Fremlin. Second printing.
  • Hewitt, Edward; Stromberg, Karl (1965). Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable. Springer. ISBN 0-387-90138-8.
  • Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer Verlag, ISBN 3-540-44085-2
  • R. Duncan Luce and Louis Narens (1987). "measurement, theory of," The New Palgrave: A Dictionary of Economics, v. 3, pp. 428–32.
  • Luther, Norman Y (1967). "A decomposition of measures". Canadian Journal of Mathematics. 20: 953–959. doi:10.4153/CJM-1968-092-0.
  • Mukherjea, A; Pothoven, K (1985). Real and Functional Analysis, Part A: Real Analysis (Second ed.). Plenum Press.
    • The first edition was published with Part B: Functional Analysis as a single volume: Mukherjea, A; Pothoven, K (1978). Real and Functional Analysis (First ed.). Plenum Press.
  • M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.
  • Nielsen, Ole A (1997). An Introduction to Integration and Measure Theory. Wiley. ISBN 0-471-59518-7.
  • 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 0-12-095780-9
  • Royden, H.L.; Fitzpatrick, P.M. (2010). Real Analysis (Fourth ed.). Prentice Hall. p. 342, Exercise 17.8. First printing. Note that there is a later (2017) second printing. Though usually there is little difference between the first and subsequent printings, in this case the second printing not only deletes from page 53 the Exercises 36, 40, 41, and 42 of Chapter 2 but also offers a (slightly, but still substantially) different presentation of part (ii) of Exercise 17.8. (The second printing's presentation of part (ii) of Exercise 17.8 (on the Luther[11] decomposition) agrees with usual presentations,[2][21] whereas the first printing's presentation provides a fresh perspective.)
  • 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.
  • Teschl, Gerald, Topics in Real and Functional Analysis, (lecture notes)
  • Tao, Terence (2011). An Introduction to Measure Theory. Providence, R.I.: American Mathematical Society. ISBN 9780821869192.
  • Weaver, Nik (2013). Measure Theory and Functional Analysis. World Scientific. ISBN 9789814508568.

ReferencesEdit

  1. ^ Fremlin, D. H. (2010), Measure Theory, vol. 2 (Second ed.), p. 221
  2. ^ a b c Mukherjea 1985, p. 90.
  3. ^ Folland 1999, p. 25.
  4. ^ Edgar 1998, Theorem 1.5.2, p. 42.
  5. ^ Edgar 1998, Theorem 1.5.3, p. 42.
  6. ^ a b Nielsen 1997, Exercise 11.30, p. 159.
  7. ^ Fremlin 2016, Section 213X, part (c).
  8. ^ Royden 2010, Exercise 17.8, p. 342.
  9. ^ Hewitt 1965, part (b) of Example 10.4, p. 127.
  10. ^ Fremlin 2016, Section 211O, p. 15.
  11. ^ a b Luther 1967, Theorem 1.
  12. ^ Mukherjea 1985, part (b) of Proposition 2.3, p. 90.
  13. ^ Fremlin 2016, part (a) of Theorem 243G, p. 159.
  14. ^ a b Fremlin 2016, Section 243K, p. 162.
  15. ^ Fremlin 2016, part (a) of the Theorem in Section 245E, p. 182.
  16. ^ Fremlin 2016, Section 245M, p. 188.
  17. ^ Berberian 1965, Theorem 39.1, p. 129.
  18. ^ Fremlin 2016, part (b) of Theorem 243G, p. 159.
  19. ^ Rao, M. M. (2012), Random and Vector Measures, Series on Multivariate Analysis, vol. 9, World Scientific, ISBN 978-981-4350-81-5, MR 2840012.
  20. ^ Bhaskara Rao, K. P. S. (1983). Theory of charges: a study of finitely additive measures. M. Bhaskara Rao. London: Academic Press. p. 35. ISBN 0-12-095780-9. OCLC 21196971.
  21. ^ Folland 1999, p. 27, Exercise 1.15.a.

External linksEdit