🚀 go-pugleaf

In <17dh96INN35s@function.mps.ohio-state.edu> edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:


>(I seem to remember seeing this, but can't remember for sure.)

>Let S be a metric space.  Write B(x,r) for the open ball with center
>x and radius r.  Let mu be a finite Borel measure on S.  For fixed
>r>0, does it follow that the function

>   x --->  mu(B(x,r))

>(from S to R) is a Borel function of x ?

Without loss of generality, assume that mu is a positive measure
(for a signed measure, use the Hahn decomposition).  Then your function
is not only Borel, it is lower semicontinuous, i.e. for any t,
{x: mu(B(x,r) > t} is open.

Suppose mu(B(x,r)) > t.  Since B(x,r) = union{B(x,r-1/n): n in positive 
integers}, mu(B(x,r-1/n)) > t for some positive integer n.  If d(x,y) < 1/n,
then B(x,r-1/n) is contained in B(y,r), so mu(B(y,r)) > t.

>Assume, if necessary, that S is separable.
>-- 
>  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
>  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
>  The Ohio State University      telephone: 614-292-0395 (Office)
>  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)

-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4