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

In this paper, we study the Frechet 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 piecewise linear continuous functions almost everywhere. Some application is included.

when the functions involved are measure preserving functions.In this paper we show that every measure preserving function f on [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.

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.Let I and 2 be probabillty measures defined on close sets B and B2, respectively.A measurable function f from B to B 2 is said to be measure preserving from (BI,I) to (B2,2) if Borel set A of B2, pl(f-l(A)) 2(A).We now state the main result of this for every section.
THEOREM 2.1.For every measure preserving function f over [0,I], there exists a sequence of piecewise linear measure preserving continuous functions converging to f almost everywhere.
To prove this theorem, we need several preliminary lemmas.The following lemma of Rlesz can be found in Royden [2].
LEMMA 2.1.Let {f be a sequence of measurable functions which converges in n measure to the function f.Then there is a subsequence {f which converges to f n i almost everywhere.
LEMMA 2.2.A measure preserving function f on [0,I] is monotone nondecreasing (nonincreasing) if and only if f(x) x (f(x) x).PROOF.Suppose f is monotone nondecreasing.Since f is measure preserving, f must be strictly increasing and The nontncreastng case can be proved similarly.
LEMMA 2.3.If f is a piecewise linear continuous function from [0,1] to [0,1], then f is measure preserving if and only if for 0 y but a finite number of values of y, m I 1, where the summation is taken over all the elements x t of x i the finite set {xi: f(x i) y} and m i is the slope of the line segment on the graph of f through the point (xt,Y).
REMARK 2.1.Those points y for whcih m i is not well-deflned are contained in the exceptional set.
PROOF.Suppose that the graph of f is made up of k line segments with k + endpoints, which are defined according to the partition on [0,I].Let ml, i k be the corrsponding slopes.Consider a y, 0 y 4 I, such that y is not the ordinate any endpoint.It is easy to see that f-l({y}) {xilf(xl y} is a finite set. of Each point (xi,Y) is an interior point of some llne segment lying on the graph of f.Let > 0 be small enough such that the interval [y,y + 6] does not contain the ordinate of any k + endpoints as its interior point.Then where f([ai,bl]) [y,y + 6] and one of the ai, b i is x i.If f is measure preserving, then m(f-l[y,y + 6]) .m([ai,bi]) T for all 0 y except for y being the I, -I ordinate of one of k + endpolnts.For an arbitrary interval [a,b]c [0,I] which does not contain ordinate of any endpoint, we have where the summation is over [xi, where YI"'''Yn 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 interior point.Now Hence f is measure preserving.
PROOF OF THEOREM 2.1.In the first part of the proof, we show that for arbitrarily small numbers 6 > 0, > 0 we can construct a measure preserving plecewise linear continuous function such that m({x: If(x) (x) > 6}) < .
Choose a natural number n, < 6 and consider the sets n E i {x: F i is the union of an infinite number of open intervals, consider the set Li {1j}j of all endpolnts of these open intervals.By the Bolzano-Welerstrass {j'}, the set of all limit points of L i, is nonempty.For a point Theorem, L i two cases can be considered EL i (i)If there exist two sequences of points in Li; one converges to from the right and the other converges to A from the left, then we construct an interval Iij (aj,bj) with aj,bj belonging to some open intervals of [0,i] -Fi, aj < < bj, bj -aj < e/(2J+In).(ii) If there exists only one sequence of points of L i converging to 'j from the right or from the left, then construct the interval Iij (,bj)with bj < /(2J+In) for the former and the interval lij (aj,) with -aj < /(2J+In) for the latter case, where aj, bj are elements of [0,I] F i.
In any of these cases, append the resulting interval to [0, I] F i) U U j lij)) is a closed set of [0,I] and is equal to the union of a finite number of closed intervals.Furthermore Thus without loss of generality, suppose that each F i has the property that F i is the union of a finit number of disjoint closed intervals of [0,I], where J=l all < bll < ai2 < hi2 < < ain i < bini, and , i l,...,n.
n J=l On F, we define a function @n(X) as follows.Restricted to each Fi, the function @n(X) is linear on each interval [alj,bij] with the absolute value of the slope equal to That is it linearly maps [a n(blj aij) It is trivial that we can extend @n to the whole interval [0,I] by adding a finite number of line segments to form a piecewise linear function h satisfying the slope condition in Lemma 2.3.Then by Lemma 2.3, n is measure preserving.Also m({x: Since I_ < n m({x: To complete the proof of the theorem, Just choose two null decreasing sequences {6 n and {e of positive numbers.For every n, we construct a measure preserving n piecewise linear function On such that m({x: If(x)-,n(X)l ) %}) < g "n It is clear that n converges to f in measure.In fact for any 6 > 0, there is a natural number n o such that for all n m({x: If(x) -n(X)l )})< m({x: If(x)-Sn(X)I )6n }) < e.
By Lemma 2.1, there is a subsequence {n k} of {n converging to the function f almost everywhere.
We remark that with a minor change in the construction of the function n in the proof of Theorem 2.1, the following result is obtained.
THEOREM 2.2.Let f be a measure preserving function over [0,1].Then there exists a sequence of one-to-one piecewlse linear measure-preserving functions over [0,I] converging to f almost everywhere.
PROOF.The proof is similar to that of Theorem 2.1.The only detail changed is the construction of n" This time over each F i we approximate the function f by a one-to-one function from F i to r[i -n I, i] and on each [aij,bij] it is linear with slope or -I.Then we extend the function to n on [0,1] by adding a finite number of line segments with slope or -I and keep the one-to-one property.It is clear that n is measure preserving since the slope condition in Lemma 2.3 is satisfied.
Theorem 2.1.has several interesting applications.One can use it to study a certain dynamic system arising from the so-called tent function (Devaney [3]) mapping from unit interval onto unit interval.To be in line with this paper we given an application arising from probability.Let B represent a probability measure on an interval B. The distribution function of the measure B is defined as F (x) (B fi (-,x]), for all x E B, and is a right continuous nondecreasing function, 0 F (x) I.If does not have any atom point, F is a continuous function on B. For an arbitrary probability measure B on B, B is completely defined if and only if F is defined.In the case of having no atom point, i.e., F is continuous, is a function from B onto [0,I].In this case, the function F is defined by F (y) inf{x: F (x) y}, for all 0 y I. Hence -X(y)]) Y m([0,y]).
COROLLARY 2.1.Let I and 2 be probability measures without atom points on closed sets B and B2, respectively.Let f be a measure preserving function from (BI,BI) to (B2,2).Then there is a sequence of measure preserving continuous functions from (BI,I) to (B2,2) that converges to f almost everywhere.

PROOF. F and F
are continuous functions on B and B2, respectively.

I 2
Of course, a corresponding corollary to Theorem 2.2 can be formulated for a measure preserving function from (BI,I) to (B2,2).
Let F be the set of all measure preserving functions from ([0,I], I) to ([0,i],2).2There is one further problem one can try to investigate.
For each i, choose a closed set F ic_ E i such n that m(F i) > m(E i) 2n n 2n and set F U F i. It is clear ian open set of [0, I], it is equal to the union of countable disjoint open intervals in [0,I].If [0,I] F i is the union of finite disjoint open intervals, F i is the union of a finite number of disjoint closed intervals.If [0,I]

F
i.It is clear that ([0, I] Fi)U( U Ii4 J is an open set of [0,I] and is the union of a finite J number of open intervals of [0, I].Then , a measure preserving function on [0,1].By Theorem 2.1, there 2 is a sequence of measure preserving piecewise linear continuous functions @n over [0,1] converging to F o f o F -1 almost everywhere.The sequence of continuous the function f almost everywhere, since F and F are continuous functions.
That is, whatare the conditions on I and 2 so that F