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Q: integration,measurable functions ( Answered 5 out of 5 stars,   1 Comment )
Subject: integration,measurable functions
Category: Science > Math
Asked by: madukar-ga
List Price: $3.00
Posted: 14 Nov 2002 12:34 PST
Expires: 14 Dec 2002 12:34 PST
Question ID: 107831
If f is a function from real line to real line is monotone, then f is
Borel measurable.


Request for Question Clarification by mcfly-ga on 14 Nov 2002 14:47 PST
Maybe I'm misunderstanding what you have written, but it sounds like a
statement rather than a question.  Could you reword to clarify your

Clarification of Question by madukar-ga on 14 Nov 2002 19:41 PST
If f : R to R is monotone, then f is Borel measurable.
Subject: Re: integration,measurable functions
Answered By: dannidin-ga on 15 Nov 2002 00:43 PST
Rated:5 out of 5 stars
Hi Madukar-ga,

First, let us review some definitions. Recall that the Borel
sigma-field (or sigma-algebra) on R is defined to be the minimal
collection of subsets of R which contains the intervals and is closed
under complementation and under countable intersection operations.
This definition contains a statement that needs to be proven, that
such a minimal collection exists: Well, take the collection which is
the intersection of all sigma-fields on R (i.e. collections of subsets
that are closed under complementation and countable intersection) that
contain the intervals; you get a sigma-field, containing the
intervals, which is obviously the minimal one having these properties,
since it is contained in any other one having these properties.

A subset of R is called a Borel set if it is in the Borel sigma-field.

Next, a function f:R->R is called Borel measurable if the inverse
image of any Borel set under f is a Borel set.

I will now prove to you that if f:R->R is monotone, then f is Borel
measurable. What we need is the following simple

Lemma: If f:R->R has the property that the inverse image of any
INTERVAL is a Borel set (compare the def. of Borel measurable
function), then f is Borel

Proof: Consider the collection C of all subsets B of R such that
(the inverse image of B under f) is a Borel subset. What do we know
about C?

1. C contains the intervals (by hypothesis)
2. C is closed under complementation: if B is in C, then f^(-1)(B) is
Borel; then f^(-1)(complement(B)) = complement(f^(-1)(B)) is also
Borel, therefore complement(B) is in C.
3. C is closed under countable intersection: if B_1, B_2, B_3, ... are
in C,
then for any n, f^(-1)(B_n) is Borel; then

f^(-1) (intersection(B_n)) = intersection(f^(-1)(B_n)) is also Borel
(since countable intersection of Borels is Borel). Therefore
intersection(B_n) is in C.

(an abbreviated way of explaining 2. and 3. above is: inverse image
preserves set operations such as complementation, intersection)

We have shown that C is a sigma-field containing the intervals.
Therefore it must contain the Borel sets, which are the minimal
sigma-field with these properties. In other words, for any Borel set
B, f^(-1)(B) is Borel. But this is exactly the definition of f being
Borel-measurable. So f is Borel measurable, Q.E.D. (for the lemma).

To finish our proof: If f is monotone, then, as mathtalk-ga indicated,
the inverse image of any interval under f is also an interval and in
particular a Borel set. Thus, by the lemma, f is Borel-measurable.
Q.E.D. (for your question)

Hope this helps. If there is anywhere in the proof where you want me
to explain in more detail, please ask!


Request for Answer Clarification by madukar-ga on 03 Dec 2002 15:15 PST

    can you prove if f is monotone then the inverse image of any
interval under f is also an interval.

Clarification of Answer by dannidin-ga on 03 Dec 2002 23:48 PST

Here's a graphical proof that the inverse image of an interval under a
monotone function is an interval:

            y axis |
                 --|            ****  
   inverse      /  |           * |
   image       /   |         **  |            *** = graph of function
   of         _|   |        *    |
   interval    |   |        *    |
  = interval   \   |       *     |
                \--|    ***      |
                   | ***|        |
                   |*   |        |
                   |--------------------> x axis

Formally, it's clear that f^ ((a,b)) = (f^(a),f^(b)) if f is monotone
increasing (f^ denotes the inverse of f) or f^((a,b)) = (f^(b),f^(a))
if f is decreasing. (f^(a),f^(b)) may be either an open, half-open, or
closed interval, depending on whether f is right-continuous at a and
left-continuous at b. But it's always an interval.

madukar-ga rated this answer:5 out of 5 stars
thank your for your kind help

Subject: Re: integration,measurable functions
From: mathtalk-ga on 14 Nov 2002 22:33 PST

Borel sets are defined in terms of intervals.  The inverse image of an
interval with respect to a monotone function is an interval.  A
function f is said to be Borel measurable if the inverse images of
Borel sets under f are again Borel sets.

regards, mathtalk

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