Exercise · ZW-HAHN-JORDAN-DECOMPOSITION-010 · ★★★ Advanced
Let \(\nu(E)=\int_E (x^2-1)\,dm\) on \([-2,2]\). Find \(\nu^+([-2,2])\) and state the Hahn set.
Choose one option.
Show answer
- A. \(\tfrac{8}{3}\)
Explanation
The Hahn set is \(\{|x|\ge 1\}\), and integrating \(x^2-1\) over \(1\le|x|\le 2\) gives \(\tfrac{8}{3}\). Forgetting the symmetry or using \([0,2]\) as the Hahn set leads to 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-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).