If $X$ is a random variable defined on $\mathbb{R}^n$ endowed with the Borel sigma-algebra and the Lebesgue measure $\lambda$. X has a density, $f$.
For $k>0$, consider the density
$$g(x_1,\ldots,x_n)=a\times f(x_1,\ldots,x_n)e^{-k\sum_{m=1}^nx_i}$$
where $a$ is a normalizing constant. $g$ defines a new random variable $Y$.
Now, could I say that the density of $Y$ under a product measure $\mu$ is $f$, where $\mu$ uses the expression $e^{-k\sum_{m=1}^nx_i}$?
Then, $\mu$ would be a product measure, right? And thus under $\mu$, is it correct that $Y$ has all the properties of $X$ under $\lambda$? Such as dependencies etc?
I am lost with these concepts of change of measure. Thanks for any pointers.