It is known that an nincreasing, leftcontinuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3increasing on $[0,\infty)^3$ but also 2decreasing in each subset of two variables (e.g. $f(x,y,z) = xyyzxz$), 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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