Hahn decomposition theorem

Let $\\mu$ be a signed measure in the measurable space $(\\Omega,\\mathscr{S})$ . There are two measurable sets $A$ and $B$ such that:

1.
$A\\cup B=\\Omega$ and $A\\cap B=\\emptyset$ ;
2.
$\\mu(E)\\geq 0$ for each $E\\in\\mathscr{S}$ such that $E\\subset A$ ;
3.
$\\mu(E)\\leq 0$ for each $E\\in\\mathscr{S}$ such that $E\\subset B$ .

The pair $(A,B)$ is called a Hahn decomposition for $\\mu$ .This decomposition is not unique, but any other such decomposition $(A^{\\prime},B^{\\prime})$ satisfies $\\mu(A^{\\prime}\\vartriangle A)=\\mu(B\\vartriangle B^{\\prime})=0$ (where $\\vartriangle$ denotes the symmetric difference), so the two decompositions differ in a set of measure 0.

单词	HahnDecompositionTheorem
释义	Hahn decomposition theorem Let $\\mu$ be a signed measure in the measurable space $(\\Omega,\\mathscr{S})$ . There are two measurable sets $A$ and $B$ such that: 1. $A\\cup B=\\Omega$ and $A\\cap B=\\emptyset$ ; 2. $\\mu(E)\\geq 0$ for each $E\\in\\mathscr{S}$ such that $E\\subset A$ ; 3. $\\mu(E)\\leq 0$ for each $E\\in\\mathscr{S}$ such that $E\\subset B$ . The pair $(A,B)$ is called a Hahn decomposition for $\\mu$ .This decomposition is not unique, but any other such decomposition $(A^{\\prime},B^{\\prime})$ satisfies $\\mu(A^{\\prime}\\vartriangle A)=\\mu(B\\vartriangle B^{\\prime})=0$ (where $\\vartriangle$ denotes the symmetric difference), so the two decompositions differ in a set of measure 0.
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