Let $(E,\mathcal E,\lambda)$ be a measure space, $I$ be a countable set and $(X_i)_{i\in I}$ be an $(E,\mathcal E)$-valued process on a probabiliyty space $(\Omega,\mathcal A,\operatorname P)$. Assume $$\mu(B):=\left|\left\{i\in I:X_i\in B\right\}\right|=\int_Bp\:{\rm d}\lambda\;\;\;\text{for all }B\in\mathcal E\tag1$$ for some density $p:E\to[0,\infty)$ on $(E,\mathcal E,\lambda)$.
How is $p$ related to the distribution of $(X_i)_{i\in I}$? In particular, can we define $p$ in a way such that $(1)$ implies that each $X_i$ has density $q$ with respect to $\lambda$, where $q$ is a given probability density on $(E,\mathcal E,\lambda)$?