Theorem. Summary-only indistinguishability [ftip-00GB]
✍️source

    Let \(r,r'\in \mathcal R\) induce the same summary law under the kernel in Definition [ftip-00GA]. Any randomized decision rule that receives only that summary has the same output law in the two worlds.

    Proof. 

      The output law is the pushforward of the common summary law through the same decision kernel. Pushforwards of equal finite measures are equal.