Exercise · ZW-HAHN-JORDAN-DECOMPOSITION-006 · ★★☆ Intermediate
Let \(\nu(E)=\int_E f\,dm\) with \(f(x)=x\) on \([-1,2]\). Find \(\nu^+([-1,2])\).
Choose one option.
Show answer
- A. \(2\)
Explanation
The Hahn set is \([0,2]\), and \(\int_0^2 x\,dx=2\). Adding the negative part or using a wrong set gives other values.
Review the concept: Hahn–Jordan decomposition
The Hahn–Jordan decomposition splits a signed measure \(\nu\) into the difference of two mutually singular positive measures, \(\nu = \nu^+ - \nu^-\).
\[\nu = \nu^+ - \nu^-,\quad \nu^+ \perp \nu^-\]
More practice problems: hahn–jordan decomposition
- ZW-HAHN-JORDAN-DECOMPOSITION-001 State what the Hahn–Jordan decomposition asserts for a signed measure nu.
- ZW-HAHN-JORDAN-DECOMPOSITION-002 State the property of the sets P and P^c in the Hahn decomposition.
- ZW-HAHN-JORDAN-DECOMPOSITION-003 Name the positive measures appearing in the Hahn–Jordan decomposition of nu.
- ZW-HAHN-JORDAN-DECOMPOSITION-004 Name the total variation of a signed measure nu in terms of nu^+ and nu^-.
- ZW-HAHN-JORDAN-DECOMPOSITION-005 State whether the Hahn–Jordan decomposition of a signed measure is unique.
- ZW-HAHN-JORDAN-DECOMPOSITION-007 With nu(E)=∫_E x dm on [-1,2], compute nu^-([-1,2]).
- ZW-HAHN-JORDAN-DECOMPOSITION-008 If nu^+(X)=3 and nu^-(X)=1, find nu(X) and |nu|(X).
- ZW-HAHN-JORDAN-DECOMPOSITION-009 Let nu=μ_1-μ_2 with μ_1 and μ_2 finite positive measures. What is nu^+ if μ_1perpμ_2 with μ_1 conce…
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