Hello there,

This is probably very easy but I can't find an argument.

Call a function $f: R^n \to R$ monotone increasing if $x_i \le y_i$ for each $1 \le i \le n$ implies $f(x) \le f(y)$.

I'd like to show that such a function is measurable; I'd be very surprised if this is not the case.

If $n=1$ it's ok, for the set $f^{-1}(-\infty,c)$ is either empty or an interval. I thought about studying each section of $f$, that is, fix e.g. $\hat{x}_1=(x_2,\ldots,x_n)$, then $f_{\hat{x}_1}(x) := f(x,x_2,\ldots,x_n)$ is monotone, hence measurable. Similarly any section is measurable. But it seems this is not sufficient to conclude...

Thanks!

Probability on Graphs. – Nate Eldredge Jun 21 '13 at 0:31