Exercise · ZW-HAHN-JORDAN-DECOMPOSITION-003 · ★☆☆ Basic
Name the positive measures appearing in the Hahn–Jordan decomposition of \(\nu\).
Choose one option.
Show answer
- D. \(\nu^+\) and \(\nu^-\)
Explanation
The two positive measures are the positive part and the negative part of the signed measure. The other options use pairs that do not form this decomposition.
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-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-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…
← ZW-HAHN-JORDAN-DECOMPOSITION-002 ZW-HAHN-JORDAN-DECOMPOSITION-004 →