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Nov 6, 2010 at 22:45 comment added R W Yes - as I said I was considering the transportation metric, but it should be true for any reasonable metric compatible with the weak$^*$ topology.
Nov 6, 2010 at 20:08 comment added gondolier Great! I am accepting it as the answer. I suppose you used the convexity in the last step since $\mu = \sum \mu(I_i) \mu_i$. BTW I have another construction which has finite support almost surely. See my own answer below.
Nov 6, 2010 at 20:07 vote accept gondolier
Nov 6, 2010 at 11:00 history answered R W CC BY-SA 2.5