9

Assume we have a measurable space $(X, \mathscr{B})$, where $X$ is a separable metric space and $\mathscr{B}$ is the Borel sigma algebra. Then, since $X$ is separable, then, $\mathscr{B}$ equals to the sigma algebra generated by open balls.

The question: assume that two probability measures on $\mathscr{B}$ are such that they agree on every open ball (just ball!) of $X$. Is it true that they are equal?

Mr.M
  • 493
  • 1
    @s.harp: None of those answers resolve this question. In $\mathbb{R}$ it's the case that every open set is a countable disjoint union of open balls, but I don't think this is true in a general separable metric space. The monotone class and $\pi$-$\lambda$ arguments don't help here because the class of open balls is not necessarily closed under finite intersections; the intersection of two open balls is open, but not necessarily a ball. – Nate Eldredge Nov 20 '16 at 15:27
  • @Nate I was too "voreilig" in that case. Here is the link anyway if it may help http://math.stackexchange.com/questions/812715/if-two-measures-agree-on-generating-sets-do-they-agree-on-all-measurable-sets/813414 – s.harp Nov 20 '16 at 15:30
  • You need at least another condition like $\sigma$-finiteness. else consider $\mathbb{R}$ with the usual metric and the counting measure resp. the counting measure restricted to $\mathbb{Q}$. But I also doubt it then, since no connected open set that is not an open ball can be written as countable disjoint union of open balls. – Dominik Nov 20 '16 at 15:54
  • @Dominik he is looking at probability measures, so counting measures are out. – s.harp Nov 20 '16 at 15:59
  • In case of $\Bbb {R}^d $ with a norm, the claim is true, see here http://math.stackexchange.com/questions/1138259/complex-measure-agreeing-on-certain-balls. I am not sure though about the general case. – PhoemueX Nov 20 '16 at 17:16
  • @s.harp The question was changed after I wrote my comment. – Dominik Nov 20 '16 at 17:50

1 Answers1

6

There is a counter-example to the general result in Theorem II of [1]. For results in the positive direction, you may be interested in [2].

References

[1] Measures not approximable or not specifiable by means of balls. Roy O. Davies, Mathematika, Volume 18, Issue 2 December 1971, pp. 157-160.

[2] Measures which agree on balls. J. Hoffmann-Jørgensen, Math. Scand. 37 (1975), no. 2, 319–326.