Let f be a function from $\displaystyle R^n $ to $\displaystyle R^+ $ such that:

Uf = {(x,y): 0 < y< f(x)} is a measurable set in $\displaystyle R^{n+1} $

The following I am copying exactly as I have it in my notes:

Suppose a > 1

Let $\displaystyle S_d $ = {(x, y-a): f(x) > a, 0 < y < (f(x) -a)/d and 0 < y < 1}

Show $\displaystyle S_d $ is measurable for all d>0

I think it's because:

$\displaystyle S_d $ is a translation Uf $\displaystyle \cap $ ((a, a+d)X$\displaystyle R^n $), which is measurable.

Any thoughts