Vietnam Journal of Mathematics 25.l (199'7\ 59-64
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6 Springer-Verlag 1997
ApproximatelyContinuousFunctionsin a MeasureSpace
B. K. Lahiri and S. Chakrabarti
Departmentof Mathematics,Uniuersityof Kalyani, Kalyani 741235,
WestBengal, India
ReceivedNovember18, 1995
Absfract. The notion of approximatelycontinuousfunctions in a measurespaceis introduced by using the conceptof density topology. Sometheoremson propertiesof such a
function are proved.
L. Introduction
Approximately continuous functions are a major source of formulating the concept of density topology, a subject of current interest to many mathematicians.
Some basic properties of approximately continuous functions may be seen in [5],
[6], [12]. A few generalizations with the domain (sometimes the range also) being
abstract spacesmay be seenin [7], [ll].
The definition of approximately continuous functions basically depends on the
structufe of open sets in the space. If we consider an arbitrary measure space, then
as such, it has no open set. But Martin [8] showed that by using the concept of the
density of a set in such a space, one can define in it a topology, known as the
density topology. Here, our purpose is to prove some basic theorems on approximately continuous functions in a measure space.
2. Preliminaries
Let (X,S,m)be a completemeasurespacein whichm(X):
outer measure m* (E) of -E is defined by
l.If E c X, then the
m- (E) : inf {m(F) : E c tr' e S}.
Let /{ be a collection [8] of sequences{f,} of sets from S such that, for each
x e X, there is at least one sequence{K"} e ff such that
(i) x e K" for all n and
(11)m(K,) --+0 as n---+@.
Any sequenceof sets satisfying (i) and (ii) is said to converge to x. The collection
of all sequencesin ff which converge to rc is denoted by tr(x).
60
B. K Lahiri and S. Chakrabarti
Assumingthe detailson the conceptof densityof setsin X [8], we note that if
ry : {E : E c X and D*(X - E,x): 0 for all x e E}
then aLis a topology, which is known as the densitytopology or d-topology on X
and the sets.E are known as d-open sets[8].
A (weak)densitytheoremis said to hold for /{ if almost all the points of every
(measurable)setis a point of the outer density(point of density)of the set.Martin
[8] observedthat a weak density theoremholds if and only if a density theorem
holdsfor /{.
A classB c S is said to cover indefinitelya set Y e X if, for eachy e Y,there
is a sequenceA, e 0 suchthat ! e Anfor all n and,m(Ar) ---+0 as n ---+@.
A family 4 = S is called m-tegdar [1], [3] if
/
\
(i) m.l U .a| . *;
\4e.il
/
(ii) the setp(A) of points outsideA and indefinitelycoveredby subsetsAI joint to
I has measutezero:
(iii) thereexistnumbersaandb (b > a> l) suchthat, for everyA e.il,
m"{o(A)} < bm(A),
where
O(A) : {OA" A' e il, A n At : A,m(A') < am(A)}.
Sincez(X) : l, condition (i) holds automatically.
The following theorem, to which we refer here as the Vitali theorem, was
provedin [3]; seealso[1].
Vitali theorem.Let A e S be a set indefinitelycoueredby the regularfamily ,il.
Thenthereexistsq sequence
{A,} of disjoint setsin .il suchthat
^ l d - o ^ ( f l r , ): lo
L
\z:1
/J
(r)
Also,for euerye > 0, thereis a sequence
{A"} of disjoint setsin ,il suchthat
f co
I
Ln:l
I
*l U A " l < m(A +
) e.
( 2)
We obtain the following corollary as an application of the Vitali theorem
which will be neededhere.
corollary A. Let e > 0 be arbitrary. under the conditionsof the vitali theorem,
thereis a positiueintegerk dependingon e suchthat
l
k
t
* l A - U t , l. ' .
L r : 1
I
Approximately
6l
Continuous Functions in a Measure Space
@
Proof. Sincethe sets{A,} aredisjoint, D*(A,)
thereis a positiveintegerk such that r:r
< oo. So, correspondingto e > 0,
m(A,)< e.
i
r:k+I
Now,
: (o-g r,)' [,^ (,y-,r')]
A- p,n,
/
@
\
/
\
-lA- U,l,)^( l) A,).
/
\ r:k+l
,/
\
r:l
So
f , o , ' l T c o ' l
' l
A , l< ^ l A- U . A +
ml l Ak- U
, I* l A- V . o , l
L t : r l L r : t l L r : k + l l
: i
m(A,), using(t)
r:kfl
<s.
I
As in [8], we will assumethroughout that a densitytheoremholds fot tr.
3. Approximately ContinuousFunctions
Let (Y, f) be a topologicalspace.By mappings,we shall alwaysmeanmappings
(functions)from X into I, unlessotherwisestated.Further, we shall assumethe
definition of measurabilityof functionsas in [4].
Definition l. A mappinsf is.said to be approximatelycontinuousat ueX if,
suchthat
for eueryGe{ withf(a)eG, thereis a set EeS with D(E,u):l
c
G.
f(E)
If / is approximatelycontinuousat each a e X, then f is called an approximately continuousfunctiorl.
Definition 2. (cf.l2l, [9], [10]) A mappingf is said to be continuouswith respectto
thedensitytopologyat x e X if, for eoeryG e { with f(x) e G, thereis a d-openset
EwithxeEandf(E)cG.
If / is continuouswith respectto the densitytopology at eachx e X, then/ is
continuouswith respectto the densitytopology'
In the real number case,the statementof Theorem I is taken as the definition
of approximatelycontinuousfunction [6], but here we needto establishthe equivalencebecausein Definition 1, a neednot be a memberof E.
B. K Lahiri and S. Chakrabarti
Theorem1. A mappingf is approximatelycontinuousif and only if it is continuous
with respectto the densitytopology.
Proof. we supposethat f is approximately continuous.Let x e x be arbitrary
and Ge7 be suchthat f(r)e G. Thereis a setEcS with D(E,x): l and
f (E) - G. lf T : E a{x}, then we observe that T is a measurable set
D(T,x) : I andf (T ) - G. By Lemma4.8 [8],
f{d -Int(r)}
:_f{y e T : D(T,y) : 1} c f(T) c G.
Thus,fI: d -rnt(T) is a d-opensetcontainingx suchthatf(H) c G.Hence,
f
is continuouswith respectto the densitytopology at x e X.
conversely,letf be continuouswith respectto the densitytopology. Let x e x
be arbitrary and G e { be suchthat /( x) e G. Thereis a d-opei set E containing
x such that f (E) c G. By corollary 4.4 and,Theorem4.5 lgl and then the density
theorem,E c ,S and,D(E,x) : l.So / is approximatelycontinuousat x.
I
Corollary. Any approximatelycontinuousfunction f : X -, y hqs the property of
Baire.
Note 1. We observethat in the present
continuity of f : X ---+I coincide.
Note 2. It follows from Theorem I that if/ is approximatelycontinuous,then it is
measurable
becauseif G e {, thenf -1(G) is d-openand so by corollary 4.4
lgl,
-'(G)
e S. However,a strongerresultis provedln the followingtheorem.
f
Theorem2. If "f is approximatelycontinuousat almost ail points of x, then is
f
measurable.
Proof.Let Ge{ and E:f-t(G).If
m.(E):0, then because
z is complete,
- s. Supposem- (E) > 0. Let y e E be a point of approximatecontinuity. th"tt
9
thereis a measurableset .86havingy as a point of densitysuchthat
f (Es) c. G,
i.e., Es c E .
By Lemma3.4[8],D(X - E,y) :0 and so by Lemma3.2in
[8],
D.(x -'E,y) < D(x - Eo,y):0.
This is true for almost all points of E. So,by Theorem4.3 in
[g], x - E e s and so
E e S. Hence,/ is measurable.
t
Theorem3. If f is measurableand (Y, {) is secondcountabre,then is approxf
imately continuousat almostall points of X.
Proof.Let {Ur,Uz,Ut,...} be a countable
basisfor I.Then M,:f-r(U,)
eS
f o r a l ln : 1 , 2 , 3 , . . . . F o r e a c hn , l e t A r : { x e M r : D ( M r , x ): l } . n y i h e ' d e n _
sitytheoremand completeness
of ra, Tr: Mn- An€ S and,m,(f):0. If y e An,
Approximately Continuous Functions in a Measure Space
63
then by Lemma 3.4 in [8],
D(A",!) : D(M" - Tn,l) : D(Mn,!) - D(T,,Y) : l.
@
Let T : U f". Then Ze S and m(T) :0.
Let X e X - T and'U e.T be such
n:l
that f(x) e U. Then there is a positiveintegerk such that f(x)eUpc. U. So,
xeMp. Sincex 47, x e,47..Also, f(At) - U7..Thus, Ap is a measurableset
having x as a point of densitysuchthat
f(Ar)-UpeU.
So,/ is approximatelycontinuousat x.
I
Combining Theorems2 and 3 we obtain the following.
Theorem4. If (Y,{) is secondcountable,then a mappingf is measurableif and
only if it is approximatelycontinuousat almostall points of X.
Note 3. Theorem 4 is a generalizationof the correspondingclassicalresult when
X : Y: R, the real numberspace[2, p.l32l. The methodof proof neededhere
has no connectionwith the classicalcase.
In the following theorem,we observethat the rangeof an approximatelycontinuous function has somespecialproperty under certain conditions(cf. t6l).
Theorem5. Let (Y, {) be a metric spacewith metric p andf : X --+Y be approximately continuous.Then f(X) is a separablesubspaceof Y, prouided that
l) Y(*) is m-regular.
xeX
Proof. Let a e X ande > 0 be arbitrary. Then thereis a setEe S with D(E, a) : I
andp(f(x), f(")) < e if x e E.
Let f[4 e tr@). There is a positiveintegerN suchthat
mlnaflql>r-e irn>N.
. *lx)''l
Thus, for all large n, p(.f (x), f (")) < e for a set of points x in Kl") whoserelative
measurewith respectto K)") is lessthan e.
Such a sequenceexists.for each a e X. Hence,by Corollary A, there existsa
suchthat
finitedisjointsequence
{K;:"} i:1,2,...,k
^l*-,Q
.,
"Jl',]
The setKr("')nu, the property that, for all y in K[!') , exceptfora setwhoserelative
64
B. K Lahiri and S. Chakrabarti
measurewith respectto KX') is lessthan s, we have
p(f (xi),f (y)) < t.
Let Br: {f (*r), f (*z),..., f (xr)}.So, exceptfor a setof measure
lessthan
'
k
,-'l
r.-/w.\r
.t,ty-t\
,
em{K}i,)}
t emltt
1 + . . . + t*txlir)} + mlx - l) xl? l
;,--,
L
r:1
k
<'f
m{KP)+ e< lm(x)-r rl : 2'
r:1
for all points q in X, f (q) is at a distance less than e from -8,.
Let {e"} be a sequence such that en ) 0, ey11< en and lim en: 0 and
€
a:
t+@
U 8.". Then -B is an enumerablesubsetof f (X) and densein f (X - M)
v:1
where M is a subset of measure zero.Let ( e M and d > 0 be arbitrary. From the
approximate continuity at (, we obtain that there is a (1 e X - M such that
pUG),f(())
< 6 1 2 . 5 o , t h e r ei s a 4 e B s u c h t h a tp ( f G ) , d
<612 andhence,
p ( f ( ( ) , q ) < 6 . 5 o , - Bi s d e n s ei n / ( X ) .
I
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