Let $(X,F,\mu)$ be a measure space, and let $f,g:X\rightarrow\mathbb{R}$ be measurable functions such that the image measures $f(\mu)$ and $g(\mu)$ are absolutely continuous with respect to one-dimensional Lebesgue measure.
Now let $h:X^2\rightarrow\mathbb{R}^2$ be defined by $h(x,y)=(f(x),g(y))$. Then my question is, is the image measure $h(\mu)$ necessarily absolutely continuous with respect to two-dimensional Lebesgue measure? If not, does anyone know of a counterexample?
Note that this question comes from me trying to better understand joint probability density functions (for non-independent random variables).
Best Answer
Not true. Take $f=g$. Let $\Delta =\{(x,x):x \in \mathbb R\}$. If $m_2$ denotes two dimensional Lebesgue measure then $m_2(\Delta)=0$. But $\mu ((f,g)^{-1} (\Delta))=1$ so $\mu (f,g)^{-1}$ is not absolutely continuous w.r.t. $m_2$.