MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is a follow up question to MO question "Idempotent measures on the free binary system?". Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law: $a * (b * c) = (a * b)* (a* c)$. Is there a finitely additive probability measure $\mu$ on $A$ such that $\mu * \mu = \mu$?

Here $\mu * \nu$ is defined by $$ \mu * \nu (X) = \int \int \mathbf{1}_{*^{-1}(X)} (x,y)\ d\mu (x) \ d \nu (y). $$ Note: because of the asymmetry of the left self distributive law, the above order of integration is important (it is possible to show that there is an idempotent with the opposite order of integration). I conjecture a positive answer (this would follow from a positive answer for the free binary system on one generator) and this question may be easier to resolve than the other.

share|cite|improve this question
And yet another bump where the original broken LaTeX was IMHO still easily parseable :-( – Yemon Choi Aug 8 '13 at 0:19

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.