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ean a measure whose range lies in 0 displaystyle 0 infty a a μ a 0 displaystyle forall a in cal a mu a in 0 infty below we give examples of 0 displaystyle 0 infty measures that are not zero measures let x displaystyle x be nonempty let a displaystyle cal a be a σ displaystyle sigma algebra on x displaystyle x let f x 0 displaystyle f x to 0 infty be not the zero function and let μ x a f x a a displaystyle mu sum _ x in a f x _ a in cal a it can be shown that μ displaystyle mu is a measure μ 0 a displaystyle mu emptyset 0 cup cal a setminus emptyset times infty 14 x 0 displaystyle x 0 a x displaystyle cal a emptyset x μ 0 x displaystyle mu emptyset 0 x infty 15 let x displaystyle x be uncountable let a displaystyle cal a be a σ displaystyle sigma algebra on x displaystyle x let c a a a is countable displaystyle cal c a in cal a a text is countable be the countable elements of a displaystyle cal a and let μ c 0 a c displaystyle mu cal c times 0 cup cal a setminus cal c times infty it can be shown that μ displaystyle mu is a measure 7 involved non example edit measures that are not semifinite are very wild when restricted to certain sets note 1 every measure is in a sense semifinite once its 0 displaystyle 0 infty part the wild part is taken away a mukherjea and k pothoven real and functional analysis part a real analysis 1985 theorem luther decomposition 16 17 for any measure μ displaystyle mu on a displaystyle cal a there exists a 0 displaystyle 0 infty measure ξ displaystyle xi on a displaystyle cal a such that μ ν ξ displaystyle mu nu xi for some semifinite measure ν displaystyle nu on a displaystyle cal a in fact among such measures ξ displaystyle xi there exists a least measure μ 0 displaystyle mu _ 0 infty also we have μ μ sf μ 0 displaystyle mu mu _ text sf mu _ 0 infty we say the 0 displaystyle mathbf 0 infty part of μ displaystyle mu to mean the measure μ 0 displaystyle mu _ 0 infty defined in the above theorem here is an explicit formula for μ 0 displaystyle mu _ 0 infty μ 0 sup μ b μ sf b b p a μ sf pre r 0 a a displaystyle mu _ 0 infty sup mu b mu _ text sf b b in cal p a cap mu _ text sf text pre mathbb r _ geq 0 _ a in cal a results regarding semifinite measures edit let f displaystyle mathbb f be r displaystyle mathbb r or c displaystyle mathbb c and let t l f μ l f 1 μ g t g f g d μ f l f 1 μ displaystyle t l_ mathbb f infty mu to left l_ mathbb f 1 mu right g mapsto t_ g left int fgd mu right _ f in l_ mathbb f 1 mu then μ displaystyle mu is semifinite if and only if t displaystyle t is injective 18 19 this result has import in the study of the dual space of l 1 l f 1 μ displaystyle l 1 l_ mathbb f 1 mu let f displaystyle mathbb f be r displaystyle mathbb r or c displaystyle mathbb c and let t displaystyle cal t be the topology of convergence in measure on l f 0 μ displaystyle l_ mathbb f 0 mu then μ displaystyle mu is semifinite if and only if t displaystyle cal t is hausdorff 20 21 johnson let x displaystyle x be a set let a displaystyle cal a be a sigma algebra on x displaystyle x let μ displaystyle mu be a measure on a displaystyle cal a let y displaystyle y be a set let b displaystyle cal b be a sigma algebra on y displaystyle y and let ν displaystyle nu be a measure on b displaystyle cal b if μ ν displaystyle mu nu are both not a 0 displaystyle 0 infty measure then both μ displaystyle mu and ν displaystyle nu are semifinite if and only if μ cld ν displaystyle mu times _ text cld nu a b μ a ν b displaystyle a times b mu a nu b for all a a displaystyle a in cal a and b b displaystyle b in cal b here μ cld ν displaystyle mu times _ text cld nu is the measure defined in theorem 39 1 in berberian 65 22 localizable measures edit localizable measures are a special case of semifinite measures and a generalization of sigma finite measures let x displaystyle x be a set let a displaystyle cal a be a sigma algebra on x displaystyle x and let μ displaystyle mu be a measure on a displaystyle cal a let f displaystyle mathbb f be r displaystyle mathbb r or c displaystyle mathbb c and let t l f μ l f 1 μ g t g f g d μ f l f 1 μ displaystyle t l_ mathbb f infty mu to left l_ mathbb f 1 mu right g mapsto t_ g left int fgd mu right _ f in l_ mathbb f 1 mu then μ displaystyle mu is localizable if and only if t displaystyle t is bijective if and only if l f μ displaystyle l_ mathbb f infty mu is l f 1 μ displaystyle l_ mathbb f 1 mu 23 19 s finite measures edit main article s finite measure a measure is said to be s finite if it is a countable sum of finite measures s finite measures are more general than sigma finite ones and have applications in the theory of stochastic processes non measurable sets edit main article non measurable set 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 edit 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 24 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 more generally see measure theory in topological vector spaces 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 displaystyle l infty 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 25 cf ba space for information about bounded charges where we say a charge is bounded to mean its range is a bounded subset of r see also edit mathematics portal abelian von neumann algebra almost everywhere carathéodory s extension theorem content measure theory fubini s theorem fatou s lemma fuzzy measure theory geometric measure theory hausdorff measure inner measure lebesgue integration lebesgue measure lorentz space lifting theory measurable cardinal measurable function minkowski content outer measure product measure pushforward measure random measure regular measure vector measure valuation measure theory volume form notes edit one way to rephrase our definition is that μ displaystyle mu is semifinite if and only if a μ pre b a 0 μ b displaystyle forall a in mu text pre infty exists b subseteq a 0 mu b infty negating this rephrasing we find that μ displaystyle mu is not semifinite if and only if a μ pre b a μ b 0 displaystyle exists a in mu text pre infty forall b subseteq a mu b in 0 infty for every such set a displaystyle a the subspace measure induced by the subspace sigma algebra induced by a displaystyle a i e the restriction of μ displaystyle mu to said subspace sigma algebra is a 0 displaystyle 0 infty measure that is not the zero measure bibliography edit robert g bartle 1995 the elements of integration and lebesgue measure wiley interscience bauer heinz 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 vladimir i 2006 measure theory berlin springer isbn 978 3540345138 bourbaki nicolas 2004 integration i springer verlag isbn 3 540 41129 1 chapter iii dudley richard m 2002 real analysis and probability cambridge university press isbn 978 0521007542 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 herbert federer 1969 geometric measure theory die grundlehren der mathematischen wissenschaften band 153 springer verlag isbn 978 3 540 60656 7 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 s2cid 124262782 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 doi 10 1007 978 1 4684 2331 0 isbn 978 1 4684 2333 4 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 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 16 decomposition agrees with usual presentations 7 26 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 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 references edit archimedes measuring the circle heath t l 1897 measurement of a circle the works of archimedes osmania university digital library of india cambridge university press pp 91 98 thomas w hawkins jr 1970 lebesgue s theory of integration its origins and development pages 66 7 university of wisconsin press isbn 0 299 05550 7 fremlin d h 2010 measure theory vol 2 second ed p 221 2 2 measures pdf heil math gatech edu archived pdf from the original on 1 july 2023 retrieved 10 march 2026 1 2 3 4 more measure theory pdf e math cornell edu archived pdf from the original on 28 february 2023 retrieved 10 march 2026 1 2 3 mukherjea pothoven 1985 p 90 folland 1999 p 25 edgar 1998 theorem 1 5 2 p 42 edgar 1998 theorem 1 5 3 p 42 1 2 nielsen 1997 exercise 11 30 p 159 fremlin 2016 section 213x part c royden fitzpatrick 2010 exercise 17 8 p 342 hewitt stromberg 1965 part b of example 10 4 p 127 fremlin 2016 section 211o p 15 1 2 luther 1967 theorem 1 mukherjea pothoven 1985 part b of proposition 2 3 p 90 fremlin 2016 part a of theorem 243g p 159 1 2 fremlin 2016 section 243k p 162 fremlin 2016 part a of the theorem in section 245e p 182 fremlin 2016 section 245m p 188 berberian 1965 theorem 39 1 p 129 fremlin 2016 part b of theorem 243g p 159 rao m m 2012 random and vector measures series on multivariate analysis vol 9 world scientific isbn 978 981 4350 81 5 mr 2840012 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 folland 1999 p 27 exercise 1 15 a external links edit look up measurable in wiktionary the free dictionary measure encyclopedia of mathematics ems press 2001 1994 tutorial measure theory for dummies v t e measure theory basic concepts absolute continuity of measures lebesgue integration l p spaces measure measure space probability space measurable space function sets almost everywhere atom baire set borel set equivalence relation borel space carathéodory s criterion cylindrical σ algebra cylinder set 𝜆 system essential range infimum supremum locally measurable π system σ algebra non measurable set vitali set null set support transverse measure universally measurable types of measures atomic baire banach besov borel brown complex complete content logarithmically convex decomposable discrete equivalent finite inner quasi invariant locally finite maximising metric outer outer perfect pre measure sub probability projection valued radon random regular borel regular inner regular outer regular saturated set function σ finite s finite signed singular spectral strictly positive tight vector particular measures counting dirac euler gaussian haar harmonic hausdorff intensity lebesgue infinite dimensional logarithmic product projections pushforward spherical tangent trivial young maps measurable function bochner strongly weakly convergence almost everywhere of measures in measure of random variables in distribution in probability cylinder set measure random compact set element measure process variable vector projection valued measure main results carathéodory s extension theorem convergence theorems dominated monotone vitali decomposition theorems hahn jordan maharam s egorov s fatou s lemma fubini s fubini tonelli hölder s inequality minkowski inequality radon nikodym riesz markov kakutani representation theorem other results disintegration theorem lifting theory lebesgue s density theorem lebesgue differentiation theorem sard s theorem vitali hahn saks theorem for lebesgue measure isoperimetric inequality brunn minkowski theorem milman s reverse minkowski steiner for...