Conditioning and compactness

For positive-probability $E$, total-variation convergence implies

For example, if $d_{\mathrm{TV}}({\mu }_n,\mu )={\eta }_n<\mu (E)/2$, the conditional distance is at most $4{\eta }_n/\mu (E)$. This is convergence of conditional laws, not a deterministic physical postselection protocol.

Conditioning a product law on equality of two independent fair bits creates correlated bits. It need not be realizable with the same disconnected support in the product-default class.

Finite outcome simplices and the hyperspace of nonempty compact subsets of a compact metric space are compact. They give subsequential support–law limits Burago. They do not, without more work:

• fill a subsequence into a family at every fine scale;

• preserve matched modifications and composition;

• provide common quantum input/output embeddings;

• establish diamond-norm convergence.

Full-tail filling is one purpose of the conditional transport contract below. It is not obtained merely by renaming all subsequential limits as experiments.