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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.

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    $\begingroup$ 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 $X$. $\endgroup$
    – Ramiro de la Vega
    Commented Feb 11, 2013 at 21:09
  • $\begingroup$ But atoms of the finite algebra are not atoms of $\mathcal A$. $\endgroup$
    – Galle
    Commented Feb 12, 2013 at 0:21
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    $\begingroup$ @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). $\endgroup$
    – Michael Greinecker
    Commented Feb 12, 2013 at 7:56

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