Relation between support of image-measure and closure of the image Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a measurable map. For a probability measure $\mathbb P$ denote by $\mu_\mathbb P$ the image measure  of $\mathbb P$ under the map $X$ and by $\text{supp}(\mu_\mathbb P)$ the support of $\mu_\mathbb P$, where the support is defined to be the smallest closed set such that its complement has $\mu_\mathbb P$-measure $0$. We know that $X(\Omega)$ is dense in $\text{supp}(\mu_\mathbb P),$ i.e. $\text{supp}(\mu_\mathbb P)\subseteq\overline{X(\Omega)}.$ Is there a characterization of probability measures $\mathbb P$ for which we have $\text{supp}(\mu_\mathbb P)=\overline{X(\Omega)}?$ For given $\Omega$ and $X$, does there always exist such a measure?

For given $\Omega$ and $X$, such a measure always exists. Let $\mathcal{B}$ be a countable basis for $\overline{X(\Omega)}$. Each element $B$ of $\mathcal{B}$ contains a point $y_B\in X(\Omega)$. Let $x_B\in X^{-1}\big(\{y_B\}\big)$. A probability measure that assigns positive probability to every element in $\{x_B:B\in\mathcal{B}\}$ has the desired property. 
