Exercise · ZW-HAHN-JORDAN-DECOMPOSITION-004 · ★☆☆ Basic
Name the total variation of a signed measure \(\nu\) in terms of \(\nu^+\) and \(\nu^-\).
Choose one option.
Show answer
- C. \(|\nu|=\nu^+ + \nu^-\)
Explanation
The total variation is the sum of both parts, not their difference or a maximum. Subtracting the parts gives the signed measure, not its absolute value.
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-005 State whether the Hahn–Jordan decomposition of a signed measure is unique.
- ZW-HAHN-JORDAN-DECOMPOSITION-006 Let nu(E)=∫_E f dm with f(x)=x on [-1,2]. Find nu^+([-1,2]).
- 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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