# Is a joint image measure of absolutely continuous image measures absolutely continuous?

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).

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. $$m_2$$.
• Thanks for your answer. By the way, is there a standard name for the relationship between $h(\mu)$ with $f(\mu)$ and $g(\mu)$? It’s not a product measure of the two measures so what is it? Commented Sep 20, 2019 at 13:43
• In probability theory it is called the joint distribution of $f$ and $g$. Commented Sep 20, 2019 at 14:12