The proof of theorem is wrong

Let’s create a very simple measurable space: $X=\\{a,b\\}$ , $\\mathcal{A}=\\{\\emptyset,\\{a\\},\\{b\\},X\\}$ .

Let’s take the $\\pi$ -system $P=\\{\\{a\\}\\}$ containing only one subset of $X$ .

Let’s create two measures $\\mu=\\delta_{a}+\\delta_{b}$ and $\u=\\delta_{a}+2\\delta_{b}$ . Then obviously $\\mu$ and $\u$ agree on $P$ and are finite, but they obviously are not equal on $\\mathcal{A}$ .

The proof, however, claims that it is sufficient if $\\mu$ and $\u$ are finite. I believe that $\\mu(X)=\u(X)$ is a necessary condition.

单词	TheProofOfTheoremIsWrong
释义	The proof of theorem is wrong Let’s create a very simple measurable space: $X=\\{a,b\\}$ , $\\mathcal{A}=\\{\\emptyset,\\{a\\},\\{b\\},X\\}$ . Let’s take the $\\pi$ -system $P=\\{\\{a\\}\\}$ containing only one subset of $X$ . Let’s create two measures $\\mu=\\delta_{a}+\\delta_{b}$ and $\u=\\delta_{a}+2\\delta_{b}$ . Then obviously $\\mu$ and $\u$ agree on $P$ and are finite, but they obviously are not equal on $\\mathcal{A}$ . The proof, however, claims that it is sufficient if $\\mu$ and $\u$ are finite. I believe that $\\mu(X)=\u(X)$ is a necessary condition.
随便看	HomotopyExtensionProperty homotopy extension property HomotopyGroups homotopy groups Homotopy-inductive types HomotopyInvariance homotopy invariance HomotopyLiftingProperty homotopy lifting property Homotopy n-types HomotopyOfMaps homotopy of maps HomotopyOfPaths homotopy of paths Homotopy theory HomotopyTypeTheory Homotopy type theory Homotopy Type Theory HomotopyWithAContractibleDomain homotopy with a contractible domain HopfAlgebra Hopf algebra HopfBifurcationTheorem Hopf bifurcation theorem HopfBundle