Rigorous definition of joint PDF and likelihood function?

Question

Let $x=(x_1,x_2,...x_n)$ be a vector of arbitrary random variables. The parameter $\theta$ in the likelihood function induces a measure $\mu_{\theta}$ on the underlying measure space. Let $\mu_{\theta}^x$ be the pushforward measure induced on the product space $R^n$ through $(x_1,x_2,...x_n)$ and $\mu_{\theta}^i$ be the pushforward measure induced on $R$ through $x_i$. If the measure $\mu_{\theta}^x$ is absolutely continuous w.r.t the product measure $\Pi \mu_{\theta}^i$, by the Radon-Nikodym theorem, there is a measurable function $p(x)$ from $R^n \to R$ such that $\mu_{\theta}^x(A)=\int_A p(x)d(\Pi \mu_{\theta}^i)$. Is this function $p(x)$ the general definition of a joint PDF for arbitrary vector of random variables and the so called likelihood function is the same thing?