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YuerWu
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Prove that there is no finite Borel measure $\mu$ such that $\mu$ null sets equal meager set

Suppose that $([0,1],B([0,1]),\mu)$ is a measure space, here $B([0,1])$ is the set of all Borel sets on [0,1], let $N_{\mu}$ be the set of all subset S of $[0,1]$ such that S is $\mu$-negligible, let $M$ be the set of all meager sets contained in $[0,1]$, I want to show that there is no finite Borel measure $\mu$ on $[0,1]$ such that $N_{\mu}=M$, how to show this? can anyone help me? thank you in advance

YuerWu
  • 415
  • 2
  • 8