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

I apologize for probably trivial question, I am far from this field.

If $\mathcal A$ is a $\sigma$-algebra of subsets of $X$ (for example Borel sets of Cantor space $2^\omega$), can I extend to $\mathcal A$ a probability measure defined on finite subalgebra of $\mathcal A$, which contains $X$ and $\emptyset$? I recall that there is the unique extension from any algebra to smallest $\sigma$-algebra containing the algebra, but I don't know if further extension is possible.

share|cite|improve this question
Can´t you just choose an element $x$ in each of the atoms of your finite subalgebra and give to this $x$ as much weight as the measure of the atom it came from? This allows you to extend the measure to the whole power set of $X$. – Ramiro de la Vega Feb 11 '13 at 21:09
But atoms of the finite algebra are not atoms of $\mathcal A$. – Galle Feb 12 '13 at 0:21
@Galle They don't have to be. The Dirac measure $\delta_x$ that assigns measure $1$ to events containing $x$ and $0$ to all others is a measure, no matter what $\sigma$-algebra you use. The measure constructed in Ramiros argument is just a positive linear combination of such measures and hence again a measure (well, one has to adap the argument if some atom has infinite measure, but the logic is the same). – Michael Greinecker Feb 12 '13 at 7:56
Thank you very much, Ramiro and Michael – Galle Feb 12 '13 at 19:08

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.