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Q: integration,measurable functions ( Answered ,   1 Comment )
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 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. r``` 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 needs?``` 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.`
 ```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 measurable. Proof: Consider the collection C of all subsets B of R such that f^(-1)(B) (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! Regards, dannidin``` Request for Answer Clarification by madukar-ga on 03 Dec 2002 15:15 PST ```hi, 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 ```Madukar-ga, 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 f of _| | * | interval | | * | = interval \ | * | \--| *** | | ***| | |* | | |--------------------> x axis (interval) 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. Cheers, dannidin```
 madukar-ga rated this answer: `thank your for your kind help`
 ```Hint: 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```