Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

It is known that an n-increasing, left-continuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3-increasing on $[0,\infty)^3$ but also 2-decreasing in each subset of two variables (e.g. $f(x,y,z) = -xy-yz-xz$), would $\mu$ then be a positive measure but negative in each subset of two variables, as in a signed measure? If so how would a Hahn decomposition look like?

share|improve this question
    
What does "3-increasing" mean? –  Michael Greinecker Jun 30 '13 at 12:03
    
    
if the function $f$ is differentiable that is equivalent to $\frac{\partial f}{\partial x_i x_j x_m} \geq 0$ –  randomsample Jun 30 '13 at 12:11
    
The 2-dimensional sections you've mentioned are of 0 Lebesgue measure (hence of $0$ $f$ measure, becuase it is ac measure wrt to Lebesgue), so they will be "ignored" by the Hahn decomposition of the $f$ measure, which will be essentially $P=[0,\infty)^{3}$. I think this is not a research-level question, as stated currently. –  Asaf Jun 30 '13 at 15:28
    
The example cited, $f(x,y,z) = -xy-yz-xz$, assigns measure zero to all $3$-rectangles. –  Gerald Edgar Jun 30 '13 at 19:27

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.