Let $\mathcal{SM}_n^+(\mathbb{R}^n)$ be the set of symmetric positive semidefinite $n\times n-$ real matrices. Denote $\mathcal{B}(\mathbb{R}^n)$ the Borel-sigma algebra on $\mathbb{R}^n$. A map $$F:\mathcal{B}(\mathbb{R}^n)\longrightarrow \mathcal{SM}_n^+(\mathbb{R}^n)$$ such that:
(i) $F(\mathbb{R}^n)=I_n$
(ii) If $(A_i)$ is a countable family of pairwise disjoint Borel sets, then $$F(\cup_iA_i)=\sum_iF(A_i)$$
my question is under which conditions there is a probability measure $\mu$ related to $F$ through $$\forall \ \ A\in \mathcal{B}(\mathbb{R}^n)\ \ F(A)=\int_A xx^Td\mu(x)\ \ \ (*)$$
help plz thx
Revised, 29 Sept 2017. From your data you can find a finite measure $\nu$ and a function $\phi: \mathbb R^n\to\mathcal{SM}_n^+(\mathbb R^n)$ such that $F(A)=\int_A \phi(x)d\nu(x)$, as shown in the next paragraph. If your $(*)$ holds, $d\mu = 1/\|x\|^2 d\nu$. The condition you seek is that $\phi(x)=xx^T/\|x\|^2$ for $\nu$-almost all $x$; only this way will $(*)$ be verified. One necessary condition is that if $A=\{x: \|x\|\le t\}$ then $\operatorname{tr}(A)\le t^2$. Others, too complicated to state here, put constraints on $F(A)$ when $A=B(x,r),$ the $r$-ball centered at $x$: then $F(A)$ must be in the cone of psd matrices generated by some neighborhood of $xx^T$.
Here is the recipe for $\nu$ and $\phi$: Let $\nu(A) = \operatorname{tr}(F(A))$. Observe that the matrix elements of $F$ define signed measures: $m_{ij}(A) = F(A)_{ij}$. It is easy to check that each $m_{ij}\ll\nu$, so the Radon Nikodym derivatives $\phi_{ij}(x) = \frac{dm_{ij}}{d\nu}$ exist, with the stated property that $F(A)=\int_A \phi(x)d\nu(x)$.