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Let $\mu$ be a signed measure. I want to prove the following:
(1) If $A$ is a positive set for $\mu$, then $\mu(A)=|\mu(A)|$.
(2) If $A$ is a negative set for $\mu$, then $\mu(A)=-|\mu(A)|$.

This is what I have done:

From $\mu^+ = \frac{1}{2}(|\mu|+\mu)$, I get that $2\mu^+(A)-\mu(A)=|\mu(A)|$.
Similarly, from $\mu^-=\frac{1}{2}(|\mu|-\mu)$, I get $2\mu^-(A)+\mu(A)=|\mu(A)$.

My questions are the following:
Can I say that $\mu^+(A)=\mu(A)$, and $\mu^-(A)=-\mu(A)$, since $A$ is positive in the first case and negative in the second? If so, how can I prove them?

  • 0
    What definition of "positive set" and "negative set" do you use that doesn't make this trivial?2012-02-02
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    $A$ is positive for $\mu$ if $A$ is measurable and every subset $B$ of $A$ is measurable and has measure greater or equal to zero. Everything stays same for negative except $\mu(B)\leq 0$.2012-02-02
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    Let $A$ be positive. Then $A$ is a measurable subset of $A$, hence $\mu(A)\geq 0$, which is equivalent to $\mu(A)=|\mu(A)|$. The negative case is similar.2012-02-02
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    Thanks...Is what I've done wrong?2012-02-02
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    Not really, but it doesn't lead anywhere. The result follows from the definition and the fact that a nonnegative number equals its absolute value and a nonpositve number its absolute value times -1. There is no need to employ the Hahn decomposition.2012-02-03
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    @MichaelGreinecker Maybe you could collect your comments in an answer.2012-02-09

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