This question and the questions it raises are extremely interesting!!! I can't help wondering about the variant of this question where the space on which the locally-finite atomic measure is defined is the real line: (-oo,oo) . And maybe instead of defining a relation in terms of [a discrete sequence of certain kinds of moves — and the limit when such a sequence converges in a certain metric]: Perhaps instead define a nice topology on (say) {locally-finite atomic measures with countable support} (from the total variation metric? the Wasserstein metric? the compact-open topology on functions with locally-finite countable support from R^oo := Prod_{k=1,oo} (-oo,oo) to the nonnegative reals [0,oo) ?) . . . and then define ~ by mu ~ nu <=> there is a continuous curve of such measures starting at mu and ending at nu. --Dan
On Apr 6, 2015, at 9:54 AM, James Propp <jamespropp@gmail.com> wrote:
I call to math-funsters' attention my recent MathOverflow post
http://mathoverflow.net/questions/202165/transitivity-of-balanced-mass-trans...
which has a math-fun-ish quality.
One thing that I know, but that I don't prove in that post (though I do mention it in one of the two affiliated posts to MathOverflow) is that if mu is the measure that gives unit mass to every integer greater than or equal to 1, and nu is the measure that gives unit mass to every integer greater than or equal to 0, then nu cannot be obtained from mu by a single balanced-transport move, or by a finite sequence of such moves, or even by an infinite sequence of such moves convergent in the total variation metric.
I pose this as a puzzle for math-fun, not only because it has a cute answer, but because I'm hoping it has more than one cute answer, and that one of you will find a different answer to this puzzle than the one that I know!
(And it would also be nice if one of you proved or disproved transitivity, but that's a separate issue.)