Measure (mathematics)

In mathematics,measure isfunction that assigns "sizes", "volumes", or "probabilities"subsets ofgiven set. The conceptimportantmathematical analysisprobability theory.

Measure theorythat branchreal analysis which investigates sigma algebras, measures, measurable functionsintegrals. It isimportanceprobabilitystatistics.

See also Lebesgue integration,Lebesgue measure

Tablecontents

1 Formal definitions 1.1 Examples 1.2 Generalizations

Formal definitions

Formally,measure μ isfunction which assignsevery element S ofgiven sigma algebra Xvalue μ(S),non-negative real number or ∞. The following properties havebe satisfied:

• The empty set has measure zero: μ({}) = 0.
• The measurecountably additive: if E₁, E₂, E₃, ...countably many pairwise disjoint setsXEtheir union, thenmeasure μ(E)equal tosum ∑μ(E_k).

If μ ismeasure onsigma algebra X, thenmembersXcalledμ-measurable sets, ormeasurable setsshort. A set Ω together withsigma algebra X on Ω andmeasure μ on Xcalledmeasure space.

The following properties can be derived fromdefinition above:

• If E₁E₂two measurable setsE₁ beingsubsetE₂, then μ(E₁) ≤ μ(E₂).
• If E₁, E₂, E₃, ...measurable setsE_n issubsetE_n+1all n, thenunion E ofsets E_nmeasurableμ(E) = lim μ(E_n).
• If E₁, E₂, E₃, ...measurable setsE_n+1 issubsetE_nall n, thenintersection E ofsets E_nmeasurable; furthermore, if at least one ofE_n has finite measure, then μ(E) = lim μ(E_n).

A measure space Ωcalled finite if μ(Ω) isfinite real number (rather than ∞). Itcalled σ-finite if Ω iscountable unionmeasurable setsfinite measure.

σ-finite measure spaces have some very nice properties; σ-finiteness can be comparedthis respectseparabletopological spaces.

A measurable set Scallednull-set if μ(S) = 0. The measure μcalled complete if every subset ofnull-setmeasurable (and then automatically itselfnull-set).

Examples

Some important measureslisted here.

• The counting measuredefined by μ(S) = numberelementsS.
• The Lebesgue measure isunique complete translation-invariant measure onsigma algebra containingintervalssR such that μ([0,1]) = 1.
• The Haar measure forlocally compact topological group isgeneralization ofLebesgue measurehassimilar uniqueness property.
• The zero measuredefined by μ(S) = 0all S.
• Every probability space gives rise tomeasure which takesvalue 1 onwhole space (and therefore takes all its values inunit interval [0,1]). Suchmeasurecalledprobability measure. See probability axioms.

Generalizations

For certain purposes, itusefulhave"measure" whose valuesnot restricted tonon-negative reals or infinity. For instance,countably additive set functionvalues in(signed) real numberscalledsigned measure, while suchfunctionvalues incomplex numberscalledcomplex measure. A measure that takes values inBanach spacecalledspectral measure; theseused mainlyfunctional analysis forspectral theorem.

Another generalization isfinitely additive measure. This issame asmeasure except that insteadrequiring countable additivity we require only finite additivity. Historically, this definition was used first, but provedbe not so useful.

The remarkable resultintegral geometry known as Hadwiger's theorem states thatspacetranslation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unionscompact convex setsRⁿ consists (upscalar multiples)one "measure" that"homogeneousdegree k"each k=0,1,2,...,n,linear combinationsthose "measures". "Homogeneousdegree k" means that rescaling any set by any factor c>0 multipliesset's "measure" by c^k. The one thathomogeneousdegree n isordinary n-dimensional volume. The one thathomogeneousdegree n-1 is"surface volume". The one thathomogeneousdegree 1 ismysterious function called"mean width",misnomer. The one thathomogenousdegree 0 isEuler characteristic.