an ultrafilter $U$ on a set $X$ is a collection of subsets of $X$ that is a filter, that cannot be enlarged (as a filter). An ultrafilter may be considered as a finitely additive measure. Then every subset of $X$ is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0). ... define a function $m$ on the power set of $X$ by setting $m(A) = 1$ if $A$ is an element of $U$ and $m(A) = 0$ otherwise. Then $m$ is a finitely additive measure on $X$, and every property of elements of $X$ is either true almost everywhere or false almost everywhere.
It says "an ultrafilter may be considered as a finitely additive measure." However, I only see how an ultrafilter induces a finitely additive measure, and don't see how a finitely additive measure induces an ultrafilter. In order for a finitely additive measure to induce an ultrafilter, I think it is not enough that for any subset $A$ of $X$, either $m(A) = 1$ and $m(X-A) = 0$ or $m(X-A) = 1$ and $m(A) = 0$, isn't it?
Thanks and regards!