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 $$\mathbb{P}_X(B)=\int_B f_X(x) \; d\mu_1\times\mu_2 \quad \forall B \in \mathcal{B}_1\otimes\mathcal{B}_2?$$