Quick question about density with respect to product measure

Question

Suppose we got 2 non indepenendent random variables $X_1:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_1,\mathcal{B}_1)$ and $X_2:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_2,\mathcal{B}_2)$ with densities with respect to $\sigma$-finite measures $\mu_1$ and $\mu_2$, respectively. Is it true that random vector $X=(X_1,X_2)$ has joint density with respect to product measure $\mu_1\times\mu_2$ even if $\mathbb{P}_X\neq\mathbb{P}_{X_1}\times\mathbb{P}_{X_2}$?

Edit: Due to misunderstanding of my question, let me explain it a little bit.

Suppose we got 2 probability spaces $(\Omega_1,\mathcal{A}_1,\mathbb{P}_1)$ and $(\Omega_2,\mathcal{A}_2,\mathbb{P}_2)$. Let $X_1:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_1,\mathcal{B}_1)$ and $X_2:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_2,\mathcal{B}_2)$ be two random variables and let $\mathbb{P}_{X_1}$ and $\mathbb{P}_{X_2}$ be their distributions (measures induced by $X_1$ and $X_2$ from $\mathbb{P}_1$ and $\mathbb{P}_2$ respectively).

Suppose $X_1$ has density $f_{X_1}$ with respect to $\mu_1$, so $$\mathbb{P}_{X_1}(B)=\int_B f_{X_1}(x)\;d\mu_1\quad \forall B\in\mathcal{B}_1 $$ and $X_2$ has density $f_{X_2}$ with respect to $\mu_2$, so $$\mathbb{P}_{X_2}(B)=\int_B f_{X_2}(x)\;d\mu_2\quad \forall B\in\mathcal{B}_2 $$.

Now consider joint probability space $(\Omega_1\times\Omega_2,\mathcal{A}_1\otimes\mathcal{A}_2,\mathbb{P})$, where $\mathbb{P}$ is a probability measure of joint experiment and a random vector $(X_1,X_2)=X:(\Omega_1\times\Omega_2,\mathcal{A}_1\otimes\mathcal{A}_2)\rightarrow(\mathcal{X}_1\times\mathcal{X}_2,\mathcal{B}_1\otimes\mathcal{B}_2)$ with it's distribution $\mathbb{P}_X$.

Is it true, that random vector $X$ has density $f_X$ with respect to $\mu_1\times\mu_2$ even though $X_1$ and $X_2$ could be non indenpendent, so $$\mathbb{P}_X(B)=\int_B f_X(x) \; d\mu_1\times\mu_2 \quad \forall B \in \mathcal{B}_1\otimes\mathcal{B}_2?$$

Answer 1

For a counterexample let

• $\Omega_1 = \Omega_2 = [0,1]$
• $\mathcal A_1 = \mathcal A_2 = \mathcal B_1 = \mathcal B_2 = \mathcal B([0,1]) \equiv \mathcal B(\mathbb R)\big|_{[0,1]}$
• $\mathbb P_1 = \mathbb P_2 = \mathop{\mathrm{Unif}}([0,1]) \equiv \lambda\big|_{[0,1]}$
• $X_1 = X_2 = \mathrm{id}_{[0,1]}$.

Clearly, $(X_1, X_2)$ takes values in $N \mathrel{:=} \{(x_1, x_2) \in [0,1]^2 : x_1 = x_2\}$.
But $(\mathbb P_1 \times \mathbb P_2)(N) = 0$, so $(X_1, X_2)$ can't have a density w.r.t to $\mathbb P_1 \times \mathbb P_2.$