# Measure induced by function

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?

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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