on frichet theorem in the set of measure preserving functions ... - EMIS

373

Internat. J. Math. & Math. Sci. VOL. 13 NO. 2 (1990) 373-378

ON FRICHET THEOREM IN THE SET OF MEASURE PRESERVING FUNCTIONS OVER THE UNIT INTERVAL

SO-HSIANG CHOU and TRUC T. NGUYEN Department of Mathematics and Statistics Bowling Green State University Bowling Green, Ohio 43403-0221 (Received November 30, 1988)

ABSTRACT.

In this paper, we study the Fr4chet theorem in the set of measure

preserving functions over the unit interval and show that any measure preserving function on

[0,1] can be approximated by a sequence of measure preserving plecewlse

lnear continuous functions almost everywhere.

KEY WORDS AND PHRASES.

Some appllcatlon is included.

Measure preserving, Borel set, distribution function, spline.

1980 AMS SUBJECT CLASSIFICATION CODE. 28A20.

I.

INTRODUCTION.

A theorem of Frchet states that for every measurable function f(x) which is defined and is finite almost everywhere on the closed interval [a,b], there exists a sequence of continuous function converging to f(x) almost everywhere (Natanson [I]). In certain applications it is important to know whether a Fr4chet-type theorem holds when the functions involved are measure preserving functions. that every measure preserving function f on

In this paper we show

[0,1] can be approximated by

a sequence of

plecewlse llnear measure preserving continuous functions almost everywhere.

More

preclsely, given a measure preserving function f on [0, I], there exists a sequence of plecewlse linear measure preserving continuous functions converging to f almost everywhere.

The next section contains the proof of this assertion.

Furthermore, we

show that every measure preserving function can be approximated by a sequence of plecewlse linear one-to-one measure preserving functions almost everywhere.

Also

included are some applications of the results.

2.

FRECHET THEOREM IN THE SET OF MEASURE PRESERVING FUNCTIONS OVER [0, I].

Let m denote the Lebesgue measure on [0,I]. Let f be a measurable function from a closed set B to B. f is said to be measure preserving on B if for each Borel set

A c_ B,

m(f-l(A))

m(A).

This notion can be further generalized.

probabillty measures defined on close sets B

and

B2,

respectively.

Let

I

and

2

A measurable

be

S. CHOU AND T.T. NGUYEN

374

to B 2 is said to be measure preserving from

function f from B

for every Borel set A of

pl(f-l(A))

B2,

2(A).

(BI, I)

to

(B2,2)

if

We now state the main result of this

section.

For every measure preserving function f over [0,I], there exists a

THEOREM 2.1.

sequence of piecewise linear measure preserving continuous functions converging to f almost everywhere. The following lemma

To prove this theorem, we need several preliminary lemmas. of Rlesz can be found in Royden [2].

be a sequence of measurable functions which converges in n which converges to f measure to the function f. Then there is a subsequence {f n i almost everywhere.

LEMMA 2.1.

Let {f

LEMMA 2.2.

A measure preserving function f on [0,I] is monotone nondecreasing

x (f(x) (nonincreasing) if and only if f(x) PROOF. Suppose f is monotone nondecreasing.

x). Since f is measure preserving, f

must be strictly increasing and

f-l(x).

m([0,f-l(x)])

m(f-l[0,x])

m([0,x])

x

The nontncreastng case can be proved similarly.

If f is a piecewise linear continuous function from [0,1] to [0,1],

LEMMA 2.3.

then f is measure preserving if and only if for 0

mI

values of y, x

i

the finite set

{xi:

f through the point

but a finite number of

y

1, where the summation is taken over all the elements x t of

y} and mi is the slope of the line segment on the graph of

f(x i)

(xt,Y).

Those points y for whcih mi is not well-deflned are contained in the

REMARK 2.1. exceptional set.

PROOF.

Suppose that the graph of f is made up of k line segments with k + i Let ml,

endpoints, which are defined according to the partition on [0,I]. be the corrsponding slopes.

of any endpoint. Each point

Let

>

(xi,Y)

Consider a y, 0

It is easy to see that

f-l({y})

{xilf(xl

y} is a finite set.

is an interior point of some llne segment lying on the graph of f.

0 be small enough such that the interval [y,y + 6] does not contain the

ordinate of any k

+

endpoints as its interior point.

f-1([y,y

O

+ 6]) x

where

k

y 4 I, such that y is not the ordinate

f([ai,bl])

.

f

i

-I

Then

[ai’bi]’

[y,y + 6] and one of the ai, bi is xi.

then

m(f-l[y,y

+ 6])

x

m([ai,bi])

if

({y})

If f is measure preserving,

FRCKET THEOREM

375

IN THE SET OF MEASURE PRESERVING FUNCTIONS

Thus

xi f

({y})

T

[

Conversely, if

-I ordinate of one of k

+

For an arbitrary interval [a,b]c [0,I] which does

endpolnts.

.

--

not contain ordinate of any endpoint, we have

m(f-l([a,b])) m(U [xi,x] x

:b-,,)/Im. [

a)

x

f([xl,x’i])

m([xi,x])

t

(b

where the summation is over

except for y being the

y

for all 0

I,

[xi,x’ i]

m([a,b])

a

b

-i|

i

such that

[a,b].

For general [a,b], the proof follows by partitioning [a,b] as U U )U [a,y [a,b]

YI"’’’Yn

[Yn,b],

[Yn-1’Yn

I)U [YI’Y2

where

are ordinates of endpolnts of llne segments lying on the graph of f.

Furthermore none of the above subintervals contains ordinates of endpolnts as an

Now

interior point.

m(f

-I

[a,b])

m(f m(f

-I -1

([a,y I)U

[a,Yl))

+

[Yn,b]) + m(f -I [Yn,b))

+ (b-

-a) +

(Yl

U

yn

m([a,b]). Hence f is measure preserving. PROOF OF THEOREM 2.1. arbitrarily small numbers 6 linear continuous function

In the first part of the proof, we show that for

>

i

{x:

(i n

I) 4

f(x)

n

i

0,

If(x)

(x)

>

6})




m(E i)

[0,I]

F

n U

(g

2n

E

n U i--I

and set F

2n

n

For each i, choose a closed set F ic_ E i such

i.

F

i.

It is clear

F i), and therefore,

i

n

< F i is an open set of [0, I], it is equal to the union of countable F i is the union of finite disjoint open disjoint open intervals in [0,I]. If [0,I] intervals, F i is the union of a finite number of disjoint closed intervals. If [0,I] Since

[0,I]

F i is the union of an infinite number of open intervals, consider the set of all endpolnts of these open intervals. By the Bolzano-Welerstrass the set of all limit points of L i, is nonempty. For a point Theorem, L

Li

{1j}j

{j’},

EL i

i two cases can be considered

from the (i)If there exist two sequences of points in Li; one converges to from the left, then we construct an interval right and the other converges to

Iij

A

aj,bj belonging to some open intervals -Fi, aj < < bj, bj -aj < e/(2J+In).

(aj,bj)

with

of

[0,i] (ii) If there exists only one sequence of points of L i converging to right or from the left, then construct the interval

from the

< /(2J+In)

(,bj)with bj

Iij

’j

for the former and the interval

(aj,)

lij

for the latter case, where

with

aj, bj

/(2J+In)

-aj


J=l

@n(X)

On F, we define a function function

the

[alj,bij]

[a

iJ’

bij]

onto the interval

thatn --ni] Nte n

t

[ (blj

m(Fi


and

implies

[ (bij 1

It is trivial that we can extend

@n

alj)
})

n

h

Also

is measure preserving.

m([O,l]

F)