According to Gleason's theorem, for any probability measure $p$ on the subspace lattice $\mathcal P(\mathcal H)$ of a separable Hilbert space $\mathcal H$ of dimension greater than 3, there is a unique positive self-adjoint trace class operator $D$ of trace 1 so that for any $E\in \mathcal P(\mathcal H)$,
$$p(E)=\mathrm{tr}(DE)\tag 1$$
and vice versa, if $D$ is any positive self-adjoint trace class operator of trace 1, then $\mathcal{P}(\mathcal H)\to \mathbb R: E\mapsto \mathrm{tr}(DE)$ defines a probability measure on $\mathcal P(\mathcal H)$.
Varadarajan asserts$^{1}$ that if a probability measure $p$, hence a positive self adjoint trace class operator $D$ of trace 1 is given on $\mathcal P(\mathcal H)$, then for any $E\in \mathcal P(\mathcal H)$ having $p(E)>0$, there is a unique probability measure $\mathcal c_E: \mathcal P(\mathcal H)\to \mathbb R: F\mapsto c_E(F)$ with the property that for any $F\le E$, $\displaystyle c_E(F)=\frac{p(F)}{p(E)}$, and $$c_E(F)=\frac{\mathrm{tr(EDEF)}}{\mathrm{tr(EDE)}}\tag 2$$ For this reason, $c_E$ can be regarded as a conditional probability measure $p(\cdot|E)$. It can be easily checked that for $F\le E$, that is, when $EF=FE=F$, $$c_E(F)=\frac{\mathrm{tr(EDEF)}}{\mathrm{tr(EDE)}}=\frac{\mathrm{tr}(DEFE)}{\mathrm{tr}(DEE)}=\frac{\mathrm{tr}(DF))}{\mathrm{tr}(DE))}=\frac{p(F)}{p(E)}$$
But how can we check the uniqueness of this $c_E$?
$^{1}$Varadarajan, Geometry of Quantum Theory Springer 1985, Second edition, p. 145