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 concentrated on A?

Exercise · ZW-HAHN-JORDAN-DECOMPOSITION-009 · ★★☆ Intermediate

Let \(\nu=\mu_1-\mu_2\) with \(\mu_1\) and \(\mu_2\) finite positive measures. What is \(\nu^+\) if \(\mu_1\perp\mu_2\) with \(\mu_1\) concentrated on \(A\)?

Choose one option.

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B. \(\nu^+=\mu_1\)

Explanation

By singularity, the positive part of \(\mu_1-\mu_2\) is \(\mu_1\), since \(\mu_1\) lives on \(A\) and \(\mu_2\) on \(A^c\). Subtracting or adding the measures does not respect that separation.

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Review the concept: Hahn–Jordan decomposition

The Hahn–Jordan decomposition splits a signed measure \(\nu\) into the difference of two mutually singular positive measures, \(\nu = \nu^+ - \nu^-\).

\[\nu = \nu^+ - \nu^-,\quad \nu^+ \perp \nu^-\]

Let \(\nu=\mu_1-\mu_2\) with \(\mu_1\) and \(\mu_2\) finite positive measures. What is \(\nu^+\) if \(\mu_1\perp\mu_2\) with \(\mu_1\) concentrated on \(A\)?

Explanation

Review the concept: Hahn–Jordan decomposition

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