I have a random variable $\xi $ with function distribution (f.d.) $G-$. Define the $p-$−dimensional random vector as

$X=({\psi}_{1}\xi ,...,{\psi}_{p}\xi )\sim F\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}(F{\textstyle \text{is f.d.}})$

Given $a>0$, consider $A=[-a,a{]}^{p}$. I want to find an expression for:

${\int}_{A}dF$

involving $G$ and the ${\psi}^{\prime}s$

My attempt is first try to find the f.d. of $X$:

$\begin{array}{rl}F({x}_{1},...,{x}_{p})& =\mathbb{P}[{\psi}_{1}\xi \le {x}_{1},...,{\psi}_{p}\xi \le {x}_{p}]\\ & =\mathbb{P}[\xi \le \frac{{x}_{1}}{{\psi}_{1}},...,\xi \le \frac{{x}_{p}}{{\psi}_{p}}]\\ & =\mathbb{P}[{\xi}_{n}\le min{\textstyle \{}\frac{{x}_{1}}{{\psi}_{1}},...,\frac{{x}_{p}}{{\psi}_{p}}{\textstyle \}}]\\ & =G{\textstyle (}min{\textstyle \{}\frac{{x}_{1}}{{\psi}_{1}},...,\frac{{x}_{p}}{{\psi}_{p}}{\textstyle \}}{\textstyle )}\end{array}$

Even with these expressions in hand, I'm having serious trouble finding an expression for ${\int}_{A}dF$.