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GEORGIAN MATHEMATICAL JOURNAL : Vol Period 2 Comma No Period 1 Comma 1 9 9 5 Comma\Noindent 45 Hyphengeorgian 5 3 MATHEMATICAL JOURNAL : Vol

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GEORGIAN MATHEMATICAL JOURNAL : Vol Period 2 Comma No Period 1 Comma 1 9 9 5 Comma\Noindent 45 Hyphengeorgian 5 3 MATHEMATICAL JOURNAL : Vol GEORGIAN MATHEMATICAL JOURNAL : Vol period 2 comma No period 1 comma 1 9 9 5 commannoindent 45 hyphenGEORGIAN 5 3 MATHEMATICAL JOURNAL : Vol . 2 , No . 1 , 1 9 9 5 , 45 − 5 3 ON .. SEPARABLE .. SUPPORTS .. OF .. BOREL MEASURES n centerlineA period KHARAZISHVILIfON nquad SEPARABLE nquad SUPPORTS nquad OF nquad BOREL MEASURES g Abstract period Some properties of Borel measures with separable sup hyphen n centerlineports are consideredfA . KHARAZISHVILI period .. In particularg comma it is proved that any sigma hyphen finite Borel measure on a Suslin line has a separable supports and from this n centerline f AbstractGEORGIAN . Some MATHEMATICAL properties JOURNAL of Borel : Vol . 2 , measures No . 1 , 1 9 9 5with , 45 - 5 separable 3 sup − g fact it is deduced commaON using SEPARABLE the continuum SUPPORTS hypothesis comma OF that any BOREL Suslin MEASURES line contains a Luzin subspace with the cardinality of the continuum period A . KHARAZISHVILI n centerlineLet E be a topologicalf ports are space considered period .. We . saynquad that theIn space particular E has the pro , it hyphen is proved that any Abstract . Some properties of Borel measures with separable sup - $ npertysigma open− parenthesis$ f i n i t S e closingg parenthesis if for every sigma hyphen finite Borel measure mu ports are considered . In particular , it is proved that any σ− finite defined in this space there Borel measure on a Suslin line has a separable supports and from this n centerlineexists a separablef Borel support measure comma on i period a Suslin e period line comma has a separable a separable closed set supports F open parenthesis and from this g fact it is deduced , using the continuum hypothesis , that any Suslin mu closing parenthesis subset E such that n centerline f fact it isline deduced contains a , Luzin using subspace the with continuum the cardinality hypothesis of the continuum , . that any Suslin g mu open parenthesis E backslashLet E be F opena topological parenthesis space mu . closing We say parenthesis that the closingspace E parenthesishas the pro = - 0 period n centerline f lineperty contains (S) if for a every Luzinσ− subspacefinite Borel with measure theµ defined cardinality in this space of therethe continuum . g Let us consider someexists examples a separable of topological support ,spaces i . e . having , a separable the property closed set F (µ) ⊂ E such that open parenthesis S closing parenthesis period n hspaceExample∗fn 1f period i l l g Let .. It is $ obvious E $ bethat a any topological separable topological space space . nquad E hasWe say that the space $E$ has the pro − the property open parenthesis S closing parenthesisµ( periodE n F (µ)) = 0: Example 2 period .. Let E be an arbitrary metric space whose topological weight nnoindent perty $( S )$ if forevery $ nsigma − $ finite Borel measure is not measurable in a wideLet sense us consider periodsome Then examples according of to topological the well hyphen spaces known having result the property $ nfrommu $ the topologicaldefined in measure this theory space the there space E has the property open parenthesis S closing paren- thesis period nnoindentExample 3exists period .. a Let separable E be the Alexandrov support compactification , i . e(S .): , of a some separable discrete closed set $ F ( topologicalnmu ) spacensubset period ThenE the $ following such that statements are equivalent : a closing parenthesis theExample space E has 1 . the propertyIt is obvious open that parenthesis any separable S closing topological parenthesis space semicolonE has n beginb closingf a l i parenthesis g n ∗g card open parenthesis E closing parenthesis is not measurable in a wide sense periodnmu (E nsetminus F( nmu ) ) = 0 . theproperty(S): nendExamplef a l i g n 4∗g period .. Let E be a Hausdorff topological space period We say that E is a Luzin space if every sigma hyphen finite diffused open parenthesis i period e period comma continuous n hspace ∗fn f i l l g LetExample us consider 2 . someLet examplesE be an arbitrary of topological metric space spaces whose topological having the property closing parenthesis Borelweight measure is not defined measurable in a wide sense . Then according to the well - known in E is identically zero period The classical Luzin set on the real line R is a Luzin n begin f a l i g n ∗g result from the topological measure theory the space E has the property (S): topological space openExample parenthesis 3 . aboutLet LuzinE be sets the Alexandrovsee comma forcompactification example comma of some .. brackleft discrete 1 brackright(S). closing parenthesis period .. One can easily nendf a l i g n ∗g topological space . Then the following statements are equivalent : check that any sigma hyphen finite Borela ) measure the space definedE has in the the property Luzin topological (S); space 1 99 1 Mathematics Subject Classificationb ) card (periodE) is not 54 H measurable 5 period in a wide sense . n hspaceKey words∗fn f and i l l g phrasesExample period 1 . Borelnquad measureIt is comma obvious support that comma any Luzin separable space comma topological Suslin space $ E $ has Example 4 . Let E be a Hausdorff topological space . We say that E property comma Suslinis a Luzin space if every σ− finite diffused ( i . e . , continuous ) Borel measure line period n begin f a l i g n ∗g defined in E is identically zero . The classical Luzin set on the real line R is a 45 Luzin topological space ( about Luzin sets see , for example , [ 1 ] ) . One the1 72 hyphen property 947 X slash ( 9 S5 slash ) 0 1 .0 0 hyphen 0 45 Dollar 7 period 5 0 slash 0 circlecopyrt-c 1 9 nendf a l i g n ∗g can easily 9 5 P l enum Publishingcheck Corporation that any σ− finite Borel measure defined in the Luzin topological space Example 2 . nquad Let $ E $1 99 be 1 Mathematics an arbitrary Subject Classification metric space. 54 H whose 5 . topological weight is not measurableKey in words a wide and phrases sense . Borel . Then measure according , support , Luzin to space the , well Suslin property− known , Suslin result from the topologicalline . measure theory the space $ E $ has the property $ ( S ) . $ 45 1 72 - 947 X =95=0100 − 045Dollar7:50=0circlecopyrt − c1995 P l enum Example 3 . nquadPublishingLet Corporation$ E $ be the Alexandrov compactification of some discrete topological space . Then the following statements are equivalent : n centerline fa) the space $E$ hasthe property $( S ) ; $ g n centerline fb ) card $ ( E ) $ is not measurable in a wide sense . g Example 4 . nquad Let $ E $ be a Hausdorff topological space . We say that $ E $ i s a Luzin space if every $ nsigma − $ finite diffused ( i . e . , continuous ) Borel measure defined in $ E $ is identically zero . The classical Luzin set on the real line $R$ is a Luzin topological space ( about Luzin sets see , for example , nquad [ 1 ] ) . nquad One can easily nnoindent check that any $ nsigma − $ finite Borel measure defined in the Luzin topological space n centerline f1 99 1 Mathematics Subject Classification . 54 H 5 . g Key words and phrases . Borel measure , support , Luzin space , Suslin property , Suslin l i n e . n centerline f45 g n hspace ∗fn f i l l g1 72 − 947X$/ 9 5 / 0 1 0 0 − 0 45 Dollar 7 . 5 0 / 0 circlecopyrt −c 1 9 9 5 $ P l enum Publishing Corporation 46 .. A period KHARAZISHVILI nnoindentE is concentrated46 nquad on aA countable . KHARAZISHVILI subset of E period The latter fact immediately implies that every Luzin space has the property open parenthesis S closing parenthesis period .. Noticennoindent here that$ the E $ is concentrated on a countable subset of $ E . $ The latter fact immediately impliestopological that space every E from Luzin Example space 3 is a has Luzin the space property if and onlyif $ the ( S ) . $ nquad Notice here that the topologicalcardinal number space card open $ E parenthesis $ from E Example closing parenthesis 3 is a is Luzin not measurable space in if a andwide sense only period if the cardinalWe shall show number that card the property $ ( open E parenthesis ) $ is S not closing measurable parenthesis in is nota wide comma sense generally . 46 A . KHARAZISHVILI speaking comma a hereE hyphenis concentrated on a countable subset of E: The latter fact immediately Weditary shall property show thatin the classthe of property all topological $ ( spaces S period ) $ is not , generally speaking , a here − ditary propertyimplies in the that class every of Luzin all space topological has the property spaces (S): . Notice here that the Example 5 period ..topological Denote .. space by .. theE from .. symbol Example .. R 3 to isthe a Luzin power space of * if .. and the only .. set if .. the of cardinalall .. real .. numbers Example 5 . nquadnumberDenote cardnquad (E) is notby measurablenquad the inn aquad wide sensesymbol . nquad $ R ^f ∗ g$ equipped with the SorgenfreyWe shall topology show that period the ..property Since the (S) topological is not , generally space R speaking to the power , a here of * - nquadis separablethe nquad commas e the t n productquad o hyphen f a l l spacenquad Rr to e a the l n powerquad ofnumbers * times R to the power of * is equipped with theditary Sorgenfrey property inthe topology class of all . topologicalnquad Since spaces the . topological space separable too period ThusExample we see that 5 . Denote by the symbol R∗ the set of all $ Rthe ^f space ∗ g R$ to the power of * times R to the power of * has the property open parenthesis S closing is separable , thereal product numbers equipped− space with $ the R ^Sorgenfreyf ∗ g n topologytimes .R Since ^f ∗the g topological$ is separable too . Thus we see that parenthesis period .. On the other∗ hand comma the space ∗ ∗ the space $R ^spacef ∗R gis n separabletimes ,R the ^f product ∗ g$ - space hasR the× R propertyis separable $( too .
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