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If two measures ν1,ν2 on a given σ-algebra F coincide on a π-system A such that σ(A)=F, then ν1,ν2 coincide on F.

As i understand, this means v1(E) = v2(E) for all E in A, and since A is a pi-system and sigma(A) = F, then the measures v1(E) = v2(E) for all E in F.

Main question: what would a concrete example of this look like? As i see it, if we have X = {1,2,3} and F = P(X), then for a pi-system A = {{1,2},{2,3},{2}}, sigma(A) = 2^X = F = P(X).

Now how do i select a random set (or sets) E? Say, if i choose E = {{1,2},{3}} in A, then what is the value of v1(E) and v2(E)? How would they coincide on F?

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  • $\begingroup$ I think there's some confusion here. If $m$ is a measure defined on the sigma algebra $F$, then $m(E)$ doesn't make sense, because $E$ is a subset of $F$, not a member of it. $\endgroup$
    – Jack M
    Oct 24, 2021 at 13:09
  • $\begingroup$ @JackM right, what would the measures v1 and v2 be of then? $\endgroup$
    – mojbius
    Oct 24, 2021 at 13:23
  • $\begingroup$ They're still just measures on $F$. All the stuff about $\pi$-systems doesn't change that. So they apply to elements of $F$, i.e. to subsets of $X$, in your notation. So $v_1(\{1, 2\})$ would make sense. $\endgroup$
    – Jack M
    Oct 24, 2021 at 13:29
  • $\begingroup$ @JackM What difference would v2({1,2}) make? $\endgroup$
    – mojbius
    Oct 24, 2021 at 14:07
  • $\begingroup$ IF $\{1, 2\}$ is in the $\pi$-system $A$, and if $v_1=v_2$ on $A$, then by definition $v_1(\{1, 2\})=v_2(\{1, 2\})$. $\endgroup$
    – Jack M
    Oct 24, 2021 at 14:15

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