non-subjective probability ${\mathsf{P}}_U$ for $U$ before $X=x$ is observed. Then, after $X=x$ is observed, the conditional distribution of $U$, which is concentrated on the set $\{u: x=a(\theta,u)\}$, where $\theta$ is the fixed true parameter value, is replaced by the original ${\mathsf{P}}_U$, i.e., one “continues to regard” $U$ as a sample from ${\mathsf{P}}_U$ after $X=x$ is observed [@dempster1963]. Therefore, despite starting with a non-subjective probability ${\mathsf{P}}_U$, the choice to replace the conditional distribution of $U$, given $X=x$, with ${\mathsf{P}}_U$ ultimately makes the fiducial probabilities subjective. This explains why fiducial inference is not valid for some assertions (and finite $n$). IM probabilities, on the other hand, are based on ${\mathsf{P}}_{\mathcal{S}}$, and the fact that there are no direct links between data and ${\mathcal{S}}$ means that the ${\mathsf{P}}_{\mathcal{S}}$-probabilities are the same—in terms of both computation and interpretation—before and after $X=x$ is observed. This explains why IM-based inference is valid whenever the predictive random set is suitably calibrated to ${\mathsf{P}}_U$; see Section \[SS:validity\] below.

Finally, without practical loss of generality, assume that $\{{\mathsf{P}}_{X|\theta}: \theta \in \Theta\}$ has a common dominating measure, say $\mu$. Then we require that ${\mathsf{bel}}_x(A;{\mathcal{S}})$ be a $\mu$-measurable function in $x$. This is easy to check in examples, but general sufficient 