Probability law under change of measure

Question

I have a question regarding the law under a change of measure.

The radon nikodym theorem says, that $E_Q(X) = E_P(ZX)$ where $Z = dQ/dP$ the change of measure. I am not interested in the expectation, but in the law.

So e.g. for the distribution, does it hold that that $Q(X \leq x) = P(ZX \leq x)$? And how would I write that in a two dimensional case? I don't think like this: $Q(X_1 \leq x_1, X_2 \leq x_2) = P(ZX_1 \leq x_1, ZX_2 \leq x_2)$

I would appreciate any help, because I am a little lost.

Barney

Answer 1

$Q(X_1 \leq x_1,X_2 \leq X_2)=\int_{\{X_1\leq x_1,X_2 \leq x_2\}} ZdP$ and this cannot be written in terms of the random variables $ZX_1$ and $ZX_2$.