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.