From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: Signed Borel Measures Date: 4 Sep 1999 13:12:10 -0500 Newsgroups: sci.math In article <7qrld9$ip8$1@taliesin.netcom.net.uk>, Noel Vaillant wrote: >> Can someone please supply me with the defintion of a regular signed borel >> measure and also include a definition of what is meant when to say such a >> measure has 'mass'. >'regular' 'signed' 'borel' 'measure': >Here is my guess: >1. 'measure' indicates it is a measure >2. 'borel' indicates that such measure is defined on the >borel sigma-algebra of some topological space. >3. 'signed' indicates that such measure is not with values in >[0,+\infty] but rather, that it is a particular case of complex >measure, with values in R >4. 'regular' indicates that given any borel set B: >m(B) = sup{m(K), K compact, K \subseteq B} >m(B) = inf{m(G), G open, B \subseteq G} This is not correct. The measure has its values in R, but is the difference of two positive measures, each of which is regular. An everywhere negative measure would fail your definition of regular. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Signed Borel Measures Date: 4 Sep 1999 16:04:42 -0400 Newsgroups: sci.math In article <7qrndq$27aq@odds.stat.purdue.edu>, Herman Rubin wrote: [above article -- djr] With one more technical restriction: The set function cannot have both (+infinity) and (-infinity) in its range. (It will then follow that one of the two positive measures whose difference is m must be bounded.) The need for this restriction is obvious, once stated. Cheers, ZVK(Slavek).